Isoperimetric inequalities in convex cylinders and cylindrically bounded convex bodies

Abstract

In this paper we consider the isoperimetric profile of convex cylinders $K\times {\mathbb {R}}^q$, where $K$ is an $m$-dimensional convex body, and of cylindrically bounded convex sets, i.e, those with a relatively compact orthogonal projection over some hyperplane of ${\mathbb {R}}^{n+1}$, asymptotic to a right convex cylinder of the form $K\times {\mathbb {R}}$, with $K\subset {\mathbb {R}}^n$. Results concerning the concavity of the isoperimetric profile, existence of isoperimetric regions, and geometric descriptions of isoperimetric regions for small and large volumes are obtained.

1 Introduction

In these notes we consider the isoperimetric problem of minimizing perimeter under a given volume constraint inside a cylindrically bounded convex body, an unbounded closed convex set $C\subset {\mathbb {R}}^{n+1}$ with interior points and relatively compact projection onto the hyperplane $x_{n+1}=0$. The perimeter considered here will be the one relative to the interior of $C$. A way to deal with this isoperimetric problem is to consider the isoperimetric profile of $C$, i.e., the function assigning to each $v>0$ the infimum of the perimeter of the sets inside $C$ of volume $v$. If this infimum is achieved for some set, this will be called an isoperimetric region. The isoperimetric profile can be understood as an optimal isoperimetric inequality on $C$.

A cylindrically bounded convex set is always included and asymptotic, in a sense to be precised later, to a convex right cylinder, a set of the form $K\times {\mathbb {R}}$, where $K\subset {\mathbb {R}}^n$ is a convex body. Here we have identified ${\mathbb {R}}^n$ with the hyperplane $x_{n+1}=0$ of ${\mathbb {R}}^{n+1}$. In this work we first consider the more general convex cylinders of the form $C=K\times {\mathbb {R}}^q$, where $K\subset {\mathbb {R}}^m$ is an arbitrary convex body with interior points, and ${\mathbb {R}}^m\times {\mathbb {R}}^q={\mathbb {R}}^{n+1}$, and prove a number of results for their isoperimetric profiles. No assumption on the regularity of $\partial C$ will be made. Existence of isoperimetric regions is obtained in Proposition 3.2 following the scheme of proof by Galli and Ritoré [4], which essentially needs a uniform local relative isoperimetric inequality [17], a doubling property on $K\times {\mathbb {R}}^q$ given in Lemma 3.1, an upper bound for the isoperimetric profile of $C$ given in (2.6), and a well-known deformation controlling the perimeter in terms of the volume. A proof of existence of isoperimetric regions in Riemannian manifolds with compact quotient under their isometry groups was previously given by Morgan [13]. Regularity results in the interior follow from Gonzalez et al. [5] and Morgan [12], but no boundary regularity result is known for general convex bodies. We also prove in Proposition 3.5 that the isoperimetric profile $I$ of a convex cylinder, as well as its power $I^{(n+1)/n}$, are concave functions of the volume, a strong result that implies the connectedness of isoperimetric regions. Further assuming $C^{2,\alpha }$ regularity of the boundary of $C$, we prove in Theorem 3.6 that, for an isoperimetric region $E\subset C$, either the closure of $\partial E\cap {{\mathrm{int}}}(C)$ is connected, or $E\subset K\times {\mathbb {R}}$ is a slab. This follows from the connectedness of isoperimetric regions and from the results by Stredulinsky and Ziemer [21]. Next we consider small and large volumes. For small volumes, following Ritoré and Vernadakis [17], we show in Theorem 3.7 that the isoperimetric profile of a convex cylinder for small volumes is asymptotic to the one of its narrowest tangent cone. As a consequence, we completely characterize the isoperimetric regions of small volumes in a convex prism, i.e, a cylinder $P\times {\mathbb {R}}^q$ based on a convex polytope $P\subset {\mathbb {R}}^m$. Indeed, we show in Theorem 3.8 that the only isoperimetric regions of sufficiently small volume inside a convex prism are geodesic balls centered at the vertices with tangent cone of the smallest possible solid angle. For large volumes, we shall assume that $C$ is a right convex cylinder, i.e., $q=1$. Adapting an argument by Duzaar and Stephen [2] to the case when $\partial K$ is not smooth, we prove in Theorem 3.9 that for large volumes the only isoperimetric regions in $K\times {\mathbb {R}}$ are the slabs $K\times I$, where $I\subset {\mathbb {R}}$ is a compact interval. The case $K\times {\mathbb {R}}^q$, with $q>1$, is more involved and will be treated in a different paper (see [18] for a proof for the Riemannian product $M\times {\mathbb {R}}^k$, where $M$ is a compact Riemannian manifold without boundary).

In the second part of this paper we apply the previous results for right convex cylinders to obtain properties of the isoperimetric profile of cylindrically bounded convex bodies. In Theorem 4.1 we show that the isoperimetric profile of a cylindrically bounded convex body $C$ approaches, when the volume grows, that of its asymptotic half-cylinder. We also show the continuity of the isoperimetric profile in Proposition 4.4. Further assuming $C^{2,\alpha }$ regularity of both the cylindrically bounded convex body $C$ and of its asymptotic cylinder, we prove the concavity of $I_C^{(n+1)/n}$ and existence of isoperimetric regions of large volume in Proposition 4.5. Our final result, Theorem 4.13, implies that translations of isoperimetric regions of unbounded volume converge in Hausdorff distance to a half-slab in the asymptotic half-cylinder. The same convergence result holds for their free boundaries, that converge in Hausdorff distance to a flat $K\times \{t\}, t\in {\mathbb {R}}^+$. Theorem 4.13 is obtained from a clearing-out result for isoperimetric regions of large volume proven in Theorem 4.9 and its main consequence, lower density estimates for isoperimetric regions of large volume given in Proposition 4.10. Such lower density bounds provide an alternative proof of Theorem 3.9, given in Corollary 4.12.

We have organized this paper into four sections. The next one contains basic preliminaries, while Sect. 3 and 4 cover the already mentioned results for cylinders and cylindrically bounded sets, respectively.

2 Preliminaries

A convex body is a compact convex set with non-empty interior. If compact is replaced by closed and unbounded, we get an unbounded convex body. We refer to Schneider’s monograph [19] for background on convex sets.

The $s$-dimensional Hausdorff measure in ${\mathbb {R}}^{n+1}$ will be denoted by $H^s$, for any $s\in \mathbb {N}$. For $E \subset C$, the relative boundary of $E$ in the interior of $C$ is $\partial _{C}E=\partial E\cap {{\mathrm{int}}}{C}$. The $(n+1)$-dimensional Hausdorff measure of $E, H^{n+1}(E)$ will be denoted by $|E|$ and referred to as the volume of $E$. Moreover, for every $x\in C$ and $r>0$ we shall define the intrinsic open ball $B_C(x,r)=B(x,r)\cap {{\mathrm{int}}}{C}$, where $B(x,r)$ denotes the open Euclidean geodesic ball centered at $x$ of radius $r$. The closure of a set $E\subset {\mathbb {R}}^{n+1}$ will be denoted by $\mathrm{cl}(E)$.

We also define the relative perimeter of $E$ in the interior of $C$ by

$$\begin{aligned} P_C(E) = \sup \left\{ \int _E{{\mathrm{div}}}\xi \, d{H}^{n+1},\quad \xi \in \Gamma _0(C) ,\quad |\xi | \le 1 \right\} , \end{aligned}$$

where $\Gamma _0(C)$ is the set of smooth vector fields with compact support in ${{\mathrm{int}}}{C}$. Observe that we are only computing the ${H}^n$-measure of $\partial E$ inside the interior of $C$. We shall say that $E$ has finite perimeter in the interior of $C$, or simply that $E\subset C$ has finite perimeter, if $P_C(E)<\infty$. We refer the reader to Maggi’s monograph [10] for background on finite perimeter sets.

If $C,C^{\prime }\subset {\mathbb {R}}^{n+1}$ are convex bodies (possible unbounded) and $f:C\rightarrow C^{\prime }$ is a Lipschitz map, then, for every $s>0$ and $E\subset C$, we get $H^s(f(E)) \le {{\mathrm{Lip}}}(f)^s\, H^s(E)$. Furthermore, $f(\partial _{C}E)=\partial _{f(C)}(f(E))$. Thus we obtain

Lemma 2.1

Let $C, C^{\prime }\subset {\mathbb {R}}^{n+1}$ be (possibly unbounded) convex bodies, and $f:C\rightarrow C^{\prime }$ a bilipschitz map. Then we have

$$\begin{aligned}&{{\mathrm{Lip}}}\left( f ^{-1}\right) ^{-n}\,P_C(E) \le P_{f(C)}(f(E)) \le {{\mathrm{Lip}}}(f)^n\, P_C(E)\nonumber \\&{{\mathrm{Lip}}}\left( f ^{-1}\right) ^{-(n+1)}\,|E| \le |f(E)| \le {{\mathrm{Lip}}}(f)^{n+1}\,|E|. \end{aligned}$$

(2.1)

Remark 2.2

If $M_ i, i=1,2,3$ are metric spaces and $f_i: M_i\rightarrow M_{i+1}, i=1,2$ are lipschitz maps, then ${{\mathrm{Lip}}}(f_2\circ f_1)\le {{\mathrm{Lip}}}(f_1){{\mathrm{Lip}}}(f_2)$. Consequently if $g: M_1\rightarrow M_2$ is a bilipschitz map, then $1\le {{\mathrm{Lip}}}(g){{\mathrm{Lip}}}\left( g ^{-1}\right)$.

Given a (possibly unbounded) convex body, we define the isoperimetric profile of $C$ by

$$\begin{aligned} I_C(v)=\inf \left\{ P_C(E) : E \subset C, |E| = v \right\} . \end{aligned}$$

(2.2)

We shall say that $E \subset C$ is an isoperimetric region if $P_C(E)=I_C(|E|)$. The renormalized isoperimetric profile of $C$ is given by

$$\begin{aligned} I_C^{(n+1)/n}. \end{aligned}$$

(2.3)

Lower semicontinuity of perimeter and standard compactness results for finite perimeter sets imply that isoperimetric regions exist in a fixed bounded subset of Euclidean space.

The known results on the regularity of isoperimetric regions are summarized in the following Lemma. One can always assume that a representative of an isoperimetric region is chosen so that it is closed, includes its points of density one and does not contain the points of density zero.

Lemma 2.3

([5, 6], [20, Thm. 2.1]) Let $C\subset {\mathbb {R}}^{n+1}$ be a (possible unbounded) convex body and $E\subset C$ an isoperimetric region. Then $\partial _C E = S_0\cup S$, where $S_0\cap S=\emptyset$ and

• (i) $S$ is an embedded $C^{\infty }$ hypersurface of constant mean curvature.

• (ii) $S_0$ is closed and $H^{s}(S_0)=0$ for any $s>n-7$. Moreover, if the boundary of $C$ is of class $C^{2,\alpha }$ then $\mathrm{cl}(\partial E\cap {{\mathrm{int}}}(C))=S\cup S_0$, where

• (iii) $S$ is an embedded $C^{2,\alpha }$ hypersurface of constant mean curvature.

• (iv) $S_0$ is closed and $H^s(S_0)=0$ for any $s>n-7$.

• (v) At points of $S \cap \partial C$, $S$ meets $\partial C$ orthogonally.

The concavity of $I_C$ and $I_C^{(n+1)/n)}$ for a convex body, [7], [11, Cor. 6.11], [17, Cor. 4.2], imply

Lemma 2.4

([17, Lemma 4.9]) Let $C\subset {\mathbb {R}}^{n+1}$ be a convex body and $0<v_0<|C|$. Then

$$\begin{aligned} I_C(v)\ge \frac{I_C(v_0)}{v_0}\,v\quad \text {and}\quad I_C(v)\ge \frac{I_C(v_0)}{v_0^{n/(n+1)}}\,v^{n/(n+1)}, \end{aligned}$$

(2.4)

for all $0\le v\le v_0$.

We also have the following uniform relative isoperimetric inequality and bounds on the volume of relative balls in convex cylinders.

Proposition 2.5

Let $C=K\times {\mathbb {R}}^q$, where $K$ is an $m$-dimensional convex body. Given $r_0>0$, there exist positive constants $M, \ell _1$, only depending on $r_0$ and $C$, and a universal positive constant $\ell _2$ so that

$$\begin{aligned} P_{\overline{B}_C(x,r)}(v)\ge M\, {\min \{v,|\overline{B}_C(x,r)|-v\}}^{n/(n+1)}, \end{aligned}$$

(2.5)

for all $x\in C, 0<r\le r_0$, and $0<v<|\overline{B}(x,r)|$, and

$$\begin{aligned} \ell _1 r^{n+1} \le |\overline{B}_C(x,r)| \le \ell _2 r^{n+1}, \end{aligned}$$

(2.6)

for any $x\in C, 0<r\le r_0$.

Proof

Since the quotient of $C$ by its isometry group is compact, the proof is reduced to that of [17, Thm. 4.12]. $\square$

Let $K\subset {\mathbb {R}}^{n+1}$ be a closed convex cone with vertex $p$ . Let $\alpha (K)=H^n(\partial {B}(p,1)\cap {{\mathrm{int}}}(K))$ be the solid angle of $K$. It is known that intrinsic geodesic balls (Euclidean balls intersected with $K$) centered at the vertex are isoperimetric regions in $K$, [9, 16], and that they are the only ones [3] for general convex cones, without any regularity assumption on the boundary. The isoperimetric profile of $K$ is given by

$$\begin{aligned} I_K(v)={\alpha (K)}^{1/(n+1)}\,(n+1)^{n/(n+1)}v^{n/(n+1)}. \end{aligned}$$

(2.7)

Consequently the isoperimetric profile of a convex cone is completely determinated by its solid angle.

We define the tangent cone $C_{p}$ of a convex body $C$ at a given boundary point $p\in \partial C$ as the closure of the set

$$\begin{aligned} \bigcup _{\lambda > 0} h_{p,\lambda } (C), \end{aligned}$$

where $h_{p,\lambda }$ is the dilation of center $p$ and factor $\lambda$. Since the quotient of the cylinder $C=K\times {\mathbb {R}}^q$ by its isometry group is compact, then adapting [17, Lemma 6.1] we get the existence of points in $\partial C$ whose tangent cones are minima of the solid angle function. By (2.7), the isoperimetric profiles of tangent cones which are minima of the solid angle function coincide. The common profile will be denoted by $I_{C_{\min }}$.

Proposition 2.6

([17, Proposition 6.2]) Let $C\subset {\mathbb {R}}^{n+1}$ be a convex body (possibly unbounded), $p\in C$ and let $H\subset {\mathbb {R}}^{n+1}$ denote the closed half-space, then

$$\begin{aligned} I_C(v)\le I_{C_p}(v)\le I_H(v), \end{aligned}$$

(2.8)

for all $0\le v\le |C|$. Moreover $I_C\le I_{C_{\min }}$.

Remark 2.7

Proposition 2.6 implies that $\overline{E}\cap \partial C\ne \emptyset$ when $E\subset C$ is an isoperimetric region since, in case $\overline{E}\cap \partial C$ is empty, then $\overline{E}$ is an Euclidean ball.

Let $C\subset {\mathbb {R}}^{n+1}$ be a closed unbounded convex set. Assume there exists a hyperplane $\Pi$ such that the orthogonal projection $\pi :{\mathbb {R}}^{n+1}\rightarrow \Pi$ takes $C$ to a bounded set, and let $K$ be the bounded convex body defined as the closure of $\pi (C)$. Then $C$ is contained in the right convex cylinder $\text {cyl}(K)$ of base $K$. Since $C$ is unbounded, it contains a half-line which is necessarily parallel to the axis of $\text {cyl}(K)$. If $C$ contains a complete line, then $C=\text {cyl}(K)$ is a cylinder [19, Lemma 1.4.2]. This implies that $C$ is either a cylinder, or is contained in a half-cylinder. We shall say that the unbounded convex body $C$ is cylindrically bounded if it is contained in a convex cylinder of bounded base, and it is not a cylinder itself.

3 Isoperimetric regions in cylinders

In this Section we consider the isoperimetric problem when the ambient space is a convex cylinder $K\times {\mathbb {R}}^q$, where $K\subset {\mathbb {R}}^m$ is a convex body. We shall assume that $m+q=n+1$. Existence of isoperimetric regions in $K\times {\mathbb {R}}^q$ can be obtained following the strategy of Galli and Ritoré for contact sub-Riemannian manifolds [4] with compact quotient under their contact isometry group. One of the basic ingredients in this strategy is the relative isoperimetric inequality in Proposition 2.5. A second one is the property that any unbounded convex body $C$ is a doubling metric space.

Lemma 3.1

Let $C\subset {\mathbb {R}}^{n+1}$ be an unbounded convex body. Then

$$\begin{aligned} |B_C(x,2r)|\le (2^{n+1}+1) |B_C(x,r)|, \end{aligned}$$

(3.1)

for any $x\in C$ and any $r>0$.

Proof

Let $x\in C, r>0$ and let $K$ denote the closed cone with vertex $x$ subtended by the closure of $\partial B_C(x,r)$. Then

$$\begin{aligned} |B_C(x,2r)|&= |B_C(x,2r)\setminus B_C(x,r)|+|B_C(x,r)| \\&\le |B_K(x,2r)\setminus B_K(x,r)|+|B_C(x,r)| \\&\le |B_K(x,2r)|+|B_C(x,r)| \\&= ^{n+1} |B_K(x,r)|+|B_C(x,r)| \\&\le (2^{n+1}+1) |B_C(x,r)|, \end{aligned}$$

as we claimed. Inequality $|B_C(x,2r)\setminus B_C(x,r)|\le |B_K(x,2r)\setminus B_K(x,r)|$ follows since, by convexity and the definition of $K, B_C(x,2r)\setminus B_C(x,r)\subset B_K(x,2r)\setminus B_K(x,r)$. $\square$

Using Lemma 3.1 and Proposition 2.6 we can show

Proposition 3.2

Consider the convex cylinder $C=K\times {\mathbb {R}}^q$, where $K\subset {\mathbb {R}}^m$ is a convex body. Then isoperimetric regions exist in $K\times {\mathbb {R}}^q$ for all volumes and they are bounded.

Proof

To follow the strategy of Galli and Ritoré [4] (see Morgan [13] for a slightly different proof for smooth Riemannian manifolds), we only need a relative isoperimetric inequality (4.16) for balls $\overline{B}_C(x,r)$ of small radius with a uniform constant; the doubling property (3.1); inequality (2.8) giving an upper bound of the isoperimetric profile; and a deformation of isoperimetric sets $E$ by finite perimeter sets $E_t$ satisfying

$$\begin{aligned} |H^n(\partial E_t\cap {{\mathrm{int}}}(C))-H^n(\partial E\cap {{\mathrm{int}}}(C))|\le M\,||E_t|-|E||, \end{aligned}$$

for small $|t|$ and some constant $M>0$ not depending in $t$, which can be obtained by deforming the regular part of the boundary of $E$ using the flow associated to a vector field with compact support.

Using all these ingredients, the proof of Theorem 6.1 in [4] applies to prove existence of isoperimetric regions in $K\times {\mathbb {R}}^q$. $\square$

Let us prove now the concavity of the isoperimetric profile of the cylinder and of its power $\tfrac{n+1}{n}$. We start by proving its continuity.

Proposition 3.3

Let $C=K\times {\mathbb {R}}^q$, where $K$ is an $m$-dimensional convex body. Then $I_C$ is non-decreasing and continuous.

Proof

Given $t>0$, the smooth map $\varphi _t:C\rightarrow C$ defined by $\varphi _t(x,y)=(x,ty), x\in K, y\in {\mathbb {R}}^q$, satisfies $|\varphi _t(E)|=t^q\,|E|$. When $t\le 1$, we also have $P_C(\varphi _t(E))\le t^{q-1}\,P_C(E)$. This implies that the isoperimetric profile is a non-decreasing function: let $v_1<v_2$, and $E\subset C$ an isoperimetric region of volume $v_2$. Let $0<t<1$ so that $|\varphi _t(E)|=v_1$. We have

$$\begin{aligned} I_C(v_1)\le P(\varphi _t(E))\le P(E)=I_C(v_2). \end{aligned}$$

This shows that $I_C$ is non-decreasing.

Let us prove now the right-continuity of $I_C$ at $v$. Consider an isoperimetric region $E$ of volume $v$. We can take a smooth vector field $Z$ with support in the regular part of the boundary of $E$ such that $\int _E{{\mathrm{div}}}Z\ne 0$. The flow $\{\varphi _t\}_{t\in {\mathbb {R}}}$ of $Z$ satisfies $(d/dt)|_{t=0} |\varphi _t(E)|\ne 0$. Using the Inverse Function Theorem we obtain a smooth family $\{E_w\}$, for $w$ near $v$, with $|E_w|=w$ and $E_v=E$. The function $f(w)=P(E_w)$ satisfies $f\ge I_C$ and $I_C(v)=f(v)$. This implies that $I_C$ is right-continuous at $v$ since, for $v_i\downarrow v$, we have

$$\begin{aligned} I_C(v)=f(v)=\lim _{i\rightarrow \infty } f(v_i)\ge \lim _{i\rightarrow \infty } I_C(v_i)\ge I_C(v), \end{aligned}$$

by the monotonicity of $I_C$.

To prove the left-continuity of $I_C$ at $v$ we take a sequence of isoperimetric regions $E_i$ with $v_i=|E_i|\uparrow v$ and we consider balls $B_i$ disjoint from $E_i$ so that $|E_i\cup B_i|=|E_i|+|B_i|$. Then $I_C(v)\le P(E_i\cup B_i)=I_C(v_i)+P(B_i)\le I_C(v)+P(B_i)$ by the monotonicity of $I_C$, and the left-continuity follows by taking limits since $\lim _{i\rightarrow \infty }P(B_i)=0$. $\square$

Lemma 3.4

Let $\{K_i\}_{i\in \mathbb {N}}$ be a sequence of $m$-dimensional convex bodies converging to a convex body $K$ in Hausdorff distance. Then $\{K_i\times {\mathbb {R}}^q\}_{i\in \mathbb {N}}$ converges to $K\times {\mathbb {R}}^q$ in lipschitz distance.

Proof

By [17, Theorem 3.4], there exists a sequence of bilipschitz maps $f_i:K_i\rightarrow K$ such that ${{\mathrm{Lip}}}(f_i),{{\mathrm{Lip}}}\left( f_i^{-1}\right) \rightarrow 1$ as $i\rightarrow \infty$. For every $i\in \mathbb {N}$, define $F_i:K_i\times {\mathbb {R}}^q \rightarrow K\times {\mathbb {R}}^q$ by

$$\begin{aligned} F_i(x,y)=(f_i(x),y),\quad (x,y)\in K_i\times {\mathbb {R}}^q. \end{aligned}$$

(3.2)

Take now $(x_1,y_1), (x_2,y_2)\in K_i\times {\mathbb {R}}^q$. We have

$$\begin{aligned} |F_i(x_1,y_1)-F_i(x_2,y_2)|^2&=|f_i(x_1)-f_i(x_2)|^2+|y_1-y_2|^2 \nonumber \\&\le \max \{{{\mathrm{Lip}}}(f_i)^2,1 \}\left( |x_1-x_2|^2+|y_1-y_2|^2\right) \nonumber \\&=\max \{{{\mathrm{Lip}}}(f_i)^2,1 \}\left| (x_1,y_1)-(x_2,y_2)\right| ^2, \end{aligned}$$

(3.3)

where $|\cdot |$ is the Euclidean norm in the suitable Euclidean space. Hence we get

$$\begin{aligned} \limsup _{i\rightarrow \infty }{{\mathrm{Lip}}}(F_i)\le 1 \end{aligned}$$

since $\lim _{i\rightarrow \infty }{{\mathrm{Lip}}}(f_i)=1$. In a similar way we find $\limsup _{i\rightarrow \infty }{{\mathrm{Lip}}}\left( F_i^{-1}\right) \le 1$. By Remark 2.2, we get ${{\mathrm{Lip}}}\left( F_i^{-1}\right) {{\mathrm{Lip}}}(F_i)\ge 1$ and the proof follows. $\square$

Proposition 3.5

Let $K\subset {\mathbb {R}}^m$ be a convex body and $C=K \times {\mathbb {R}}^q$. Then $I_C^{(n+1)/n}$ is a concave function. This implies that $I_C$ is concave and every isoperimetric set in $C$ is connected.

Proof

When the boundary of a convex cylinder $C$ is smooth, its isoperimetric profile $I_C$ and its power $I_C^{(n+1)/n}$ are known to be concave using a suitable deformation of an isoperimetric region and the first and second variations of perimeter and volume, as in Kuwert [7].

By approximation [19], there exists a sequence $\{K_i\}_{i\in \mathbb {N}}$ of convex bodies in ${\mathbb {R}}^m$ with $C^\infty$ boundary such that $K_i\rightarrow K$ in Hausdorff distance. Set $C_i=K_i\times {\mathbb {R}}^q$. By Lemma 3.4, $C_i\rightarrow C$ in lipschitz distance. Fix now some $v>0$. By Proposition 3.2, there is a sequence of isoperimetric sets $E_i\subset C_i$ of volume $v$. Thus arguing as in [17, Theorem 4.1], using the continuity of the isoperimetric profile $I_C$, we get

$$\begin{aligned} I_C(v)\le \liminf _{i\rightarrow \infty } I_{C_i}(v). \end{aligned}$$

Again by Proposition 3.2 there exists an isoperimetric set $E\subset C$ of volume $v$. Arguing again as in [17, Theorem 4.1], we obtain

$$\begin{aligned} I_C(v)\ge \limsup _{i\rightarrow \infty } I_{C_i}(v). \end{aligned}$$

Combining both inequalities we get

$$\begin{aligned} I_C(v)=\lim _{i\rightarrow \infty } I_{C_i}(v). \end{aligned}$$

So $I_C^{(n+1)/n}, I_C$ are concave functions as they are pointwise limits of concave functions.

Connectedness of isoperimetric regions is a consequence of the concavity of $I_C^{(n+1)/n}$ as in [17, Theorem 4.6]. $\square$

Assume now that the cylinder $C=K\times {\mathbb {R}}^q$ has $C^{2,\alpha }$ boundary. By Theorem 2.6 in Stredulinsky and Ziemer [21], a local minimizer of perimeter under a volume constraint has the property that either $\mathrm{cl}(\partial E\cap {{\mathrm{int}}}(C))$, the closure of $\partial E\cap {{\mathrm{int}}}(C)$, is either connected or it consists of a union of parallel (totally geodesic) components meeting $\partial C$ orthogonally with the part of $C$ lying between any two of such components consisting of a right cylinder. By the connectedness of isoperimetric regions proven in Proposition 3.5, $E$ must be a slab in $K\times {\mathbb {R}}$. So we have proven the following

Theorem 3.6

Let $C=K\times {\mathbb {R}}^q$ be a convex cylinder with $C^{2,\alpha }$ boundary, and $E\subset C$ an isoperimetric region. Then either the closure of $\partial E\cap {{\mathrm{int}}}(C)$ is connected or $E$ is an slab in $K\times {\mathbb {R}}$.

Let us consider now the isoperimetric profile for small volumes. The following is inspired by [17, Theorem 6.6], although we have simplified the proof.

Theorem 3.7

Let $C=K\times {\mathbb {R}}^q$, where $K\subset {\mathbb {R}}^m$ is a convex body. Then isoperimetric regions of small volume are close to points with the narrowest tangent cone. Furthermore,

$$\begin{aligned} \lim _{v\rightarrow 0}\frac{I_C(v)}{I_{C_{\min }}(v)}=1. \end{aligned}$$

(3.4)

Proof

To prove (3.4), consider a sequence $\{E_i\}_{i\in \mathbb {N}}\subset C$ of isoperimetric regions of volumes $v_i\rightarrow 0$. By Proposition 3.5, the sets $E_i$ are connected. The key of the proof is to show

$$\begin{aligned} {{\mathrm{diam}}}(E_i)\rightarrow 0. \end{aligned}$$

(3.5)

To accomplish this we consider $\lambda _i\rightarrow \infty$ so that the isoperimetric regions $\lambda _iE\subset \lambda _i C$ have volume 1. Then we argue exactly as in [17, Theorem 6.6]. We first produce an elimination Lemma as in [17, Theorem 5.5], with $\varepsilon >0$ independent of $\lambda _i$, that yields a perimeter lower density bound [17, Corollary 5.8] independent of $\lambda _i$. Hence the sequence $\{{{\mathrm{diam}}}(\lambda _i E_i)\}_{i\in \mathbb {N}}$ must be bounded, since otherwise applying the perimeter lower density bound we would get $P_{\lambda _i C}(\lambda _i E_i)\rightarrow \infty$, contradicting Proposition 2.6. Since $\{{{\mathrm{diam}}}(\lambda _i E_i)\}_{i\in \mathbb {N}}$ is bounded, (3.5) follows.

Translating each set of the sequence $\{E_i\}_{i\in \mathbb {N}}$, and eventually $C$, we may assume that $E_i$ converges to $0\in \partial K\times {\mathbb {R}}^k$ in Hausdorff distance. Taking $r_i=({{\mathrm{diam}}}(E_i))^{1/2}$ we have ${{\mathrm{diam}}}(r_i^{-1}E_i)\rightarrow 0$ and so

$$\begin{aligned} r_i^{-1}E_i\rightarrow 0\quad \text {in Hausdorff distance}. \end{aligned}$$

(3.6)

Let $q\in {{\mathrm{int}}}(K\cap \overline{D}(0,1))$ and let $D_q$ be an $m$-dimensional closed ball centered at $q$ and contained in ${{\mathrm{int}}}(K\cap \overline{D}(0,1))$. As the sequence $r_i^{-1}K\cap \overline{D}(0,1)$ converges to $K_0\cap \overline{D}(0,1)$ in Hausdorff distance, we construct, using [17, Thm. 3.4], a family of bilipschitz maps $f_i:r_i^{-1}K\cap \overline{D}(0,1)\rightarrow K_0\cap \overline{B}(0,1)$ with $\text {Lip}(f_i), \text {Lip}\left( f_i^{-1}\right) \rightarrow 1$, where $f_i$ is the identity on $D_q$ and is extended linearly along the segments leaving from $q$. We define, as in Lemma 3.4, the maps $F_i:\left( r_i^{-1}K\cap \overline{D}(0,1)\right) \times {\mathbb {R}}^k\rightarrow \left( K_0\cap \overline{D}(0,1)\right) \times {\mathbb {R}}^k$ by $F_i(x,y)=(f_i(x),y)$. These maps satisfy ${{\mathrm{Lip}}}(F_i),\,{{\mathrm{Lip}}}\left( F_i^{-1}\right) \rightarrow 1$. Since (3.6) holds, the maps $F_i$ have the additional property

$$\begin{aligned} P_{C_0}(F_i(r_i^{-1}E_i))=P_{C_0\cap \overline{B}(0,1)}(F_i(r_i^{-1}E_i)),\quad \text {for large}\,\ i\in \mathbb {N}. \end{aligned}$$

(3.7)

Thus by Lemma 2.1 and (2.7) we get

$$\begin{aligned} \frac{P_{C}(E_i)}{|E_i|^{n/(n+1)}}&=\frac{P_{r_i^{-1}C}\left( r_i^{-1}E_i\right) }{|r_i^{-1}E_i|^{n/(n+1)}} \nonumber \\&\ge \frac{P_{C_0}\left( F_i(r_i^{-1}E_i)\right) }{|F_i\left( r_i^{-1}E_i\right) |^{n/(n+1)}}\,\left( {{\mathrm{Lip}}}(F_i){{\mathrm{Lip}}}\left( F_i^{-1}\right) \right) ^{-n} \nonumber \\&\ge {\alpha (C_0)}^{1/(n+1)}\,(n+1)^{n/(n+1)}\,\left( {{\mathrm{Lip}}}(F_i){{\mathrm{Lip}}}\left( F_i^{-1}\right) \right) ^{-n} \end{aligned}$$

(3.8)

Since $E_i$ are isoperimetric regions of volumes $v_i$, passing to the limit we get

$$\begin{aligned} \liminf _{i\rightarrow \infty }\frac{I_C(v_i)}{v_i^{n/(n+1)}}\ge {\alpha (C_0)}^{1/(n+1)}\,(n+1)^{n/(n+1)}. \end{aligned}$$

From (2.7) we obtain,

$$\begin{aligned} \liminf _{i\rightarrow \infty }\frac{I_C(v_i)}{ I_{ C_0}(v_i)}\ge 1. \end{aligned}$$

Combining this with (2.8) and the minimal property of $I_{C_{\min }}$ we deduce

$$\begin{aligned} \limsup _{i\rightarrow \infty }\frac{I_C(v_i)}{ I_{ C_0}(v_i)}\le \limsup _{i\rightarrow \infty }\frac{I_C(v_i)}{I_{C_{\min }}(v_i)}\le 1\le \liminf _{i\rightarrow \infty }\frac{I_C(v_i)}{ I_{ C_0}(v_i)} . \end{aligned}$$

Thus

$$\begin{aligned} \lim _{i\rightarrow \infty }\frac{I_C(v_i)}{ I_{C_{\min }}(v_i)}=1. \end{aligned}$$

(3.9)

By (2.7), we conclude that $C_0$ has minimum solid angle. $\square$

A convex prism $\Pi$ is a set of the form $P\times {\mathbb {R}}^q$ where $P\subset {\mathbb {R}}^m$ is a polytope. For convex prisms we are able to characterize the isoperimetric regions for small volumes.

Theorem 3.8

Let $\Pi \subset {\mathbb {R}}^{n+1}$ be a convex prism. For small volumes the isoperimetric regions in $\Pi$ are geodesic balls centered at vertices with the smallest solid angle.

Proof

Let $\{E_i\}_{i\in \mathbb {N}}$ be a sequence of isoperimetric regions in $\Pi$ with $|E_i|\rightarrow 0$. By Theorem 3.7, after translation, a subsequence of $E_i$ is close to some vertex $x$ in $\Pi$. Since ${{\mathrm{diam}}}(E_i)\rightarrow 0$ we can assume that the sets $E_i$ are also subsets of the tangent cone $\Pi _x$ and they are isoperimetric regions in $\Pi _x$. By [3] the only isoperimetric regions in this cone are, after translation, the geodesic balls centered at $x$. These geodesic balls are also subsets of $\Pi$. $\square$

To end this section, let us characterize the isoperimetric regions for large volume in the right cylinder $K\times {\mathbb {R}}$. We closely follow the proof by Duzaar and Steffen [2], which is slightly simplified by the use of Steiner symmetrization. The case of the cylinder $K\times {\mathbb {R}}^q$, with $q>1$, is more involved and will be treated in a different paper.

We shall say that a set $E\subset K\times {\mathbb {R}}$ is normalized if, for every $x\in K$, the intersection $E\cap (\{x\}\times {\mathbb {R}})$ is a segment with midpoint $(x,0)$.

Theorem 3.9

Let $C=K\times {\mathbb {R}}$, where $K\subset {\mathbb {R}}^n$ is a convex body. Then there is a constant $v_0>0$ so that the slabs $K\times I$, where $I\subset {\mathbb {R}}$ is a compact interval, are the only isoperimetric regions of volume larger than or equal to $v_0$. In particular, $I_C(v)=2H^n(K)$ for all $v\ge v_0$.

Proof

The proof is modeled on [2, Prop 2.11]. By comparison with slabs we have $I_C(v)\le 2\,H^n(K)$ for all $v>v_0$.

Let us assume first that $E\subset K\times {\mathbb {R}}$ is a normalized set of finite volume and $H^n(\partial _C E)\le 2\,H^n(K)$, and let $E^*$ be its orthogonal projection over $K_0=K\times \{0\}$. We claim that, it $H^n(K_0\setminus E^*)>0$, then there is a constant $c>0$ so that

$$\begin{aligned} H^n(\partial _C E)\ge c |E|. \end{aligned}$$

(3.10)

For $t\in {\mathbb {R}}$, we define $E_t=E\cap (K\times \{t\})$. As $E$ is normalized, we can choose $\tau >0$ so that $H^n(E_t)\le H^n(K)/2$ for $t\ge \tau$ and $H^n(E_t)> H^n(K)/2$ for $0<t<\tau$.

For $t\ge \tau$ we apply the coarea formula and Lemma 2.4 to get

$$\begin{aligned} H^n(\partial _C E)&\ge H^n(\partial _C E\cap (K\times [t,\infty )) \nonumber \\&\ge \int _{\tau }^{+\infty }H^{n-1} (\partial _C{E_s})\,ds\ge c_1\int _\tau ^{+\infty } H^n(E_s)\,ds\ge c_1|E\cap (K\times [\tau ,+\infty ))|,\nonumber \\ \end{aligned}$$

(3.11)

where $c_1$ is a constant only depending on $H^n(K)/2$.

Let $S_t=K\times \{t\}$. For $0<t<\tau$ we have

$$\begin{aligned} H^{n} (S_t \setminus E_t)\ge H^{n} (\partial _C E \cap (K\times (0,t))), \end{aligned}$$

(3.12)

since otherwise

$$\begin{aligned} H^n(K)&= H^{n} (S_t\setminus E_t)+ H^{n} (E_t) \\&< H^{n} (\partial _C E \cap (K\times (0,t)))+ H^{n} (\partial _C E \cap (K\times [t,+\infty ))) \\&\le H^{n} (\partial _C E)/2, \end{aligned}$$

and we should get a contradiction to our assumption $H^n(\partial _C E)\le 2\,H^n(K)$, what proves (3.12). So we obtain from (3.12) and Lemma 2.4

$$\begin{aligned} H^{n} (S_t\setminus E_t)&\ge H^{n} (\partial _C E \cap (K\times (0,t))) \nonumber \\&\ge \int _0^{t}H^{n-1} (\partial _C{E\cap S_t})dt \nonumber \\&\ge c_2 \int _0^{\tau }H^{n} (S_t \setminus E_t)^{(n-1)/n}dt, \end{aligned}$$

(3.13)

where $c_2$ is a constant only depending on $H^n(K)/2$. Letting $y(t)=H^{n} (S_t \setminus E_t)$, inequality (3.13) can be rewritten as the integral inequality

$$\begin{aligned} y(t)\ge c_2\int _0^{t}y(s)^{(n-1)/n}ds. \end{aligned}$$

Since $H^n(K_0\setminus E^*)>0$ by assumption and $E$ is normalized, we have $y(t)>0$ for all $t>0$, and so

$$\begin{aligned} 2\,H^n(K)\ge H^n(S_\tau \setminus E_\tau )=y(\tau )\ge \frac{c_2^n}{n^n}\,\tau ^n, \end{aligned}$$

what implies

$$\begin{aligned} \tau \le \frac{n}{c_2\,(2\,H^n(K))^{1/n}}. \end{aligned}$$

(3.14)

We finally estimate

$$\begin{aligned} |E \cap (K\times {[0,\tau ])}|=\int _0^{\tau }H^{n} (E_t)\,dt \le 2 H^{n} ({E_0})\,\tau \le \frac{n}{c_2\,(2\,H^n(K))^{1/n}} H^{n} (\partial _C E ). \end{aligned}$$

(3.15)

Combining (3.11) and (3.15), we get (3.10). This proves the claim.

Let now $E\subset K\times {\mathbb {R}}$ be an isoperimetric region of large enough volume $v$. Following Talenti [22] or Maggi [10], we may consider its Steiner symmetrized $\text {sym}\,E$. The set $\text {sym}\,E$ is normalized and we have $|E|=|\text {sym}\,E|$ and $P_C(\text {sym}\,E)\le P_C(E)$. Of course, since $E$ is an isoperimetric region we have $P_C(\text {sym}\,E)=P_C(E)$. If $H^n(K_0\setminus E^*)>0$, then (3.10) implies

$$\begin{aligned} P_C(E)=P_C(\text {sym}\,E)=H^n(\partial _C(\text {sym}\,E))\ge c\,|\text {sym}\,E|=c\,|E|, \end{aligned}$$

providing a contradiction since $I_C\le 2\,H^n(K)$.

We conclude that $H^n(K_0\setminus E^*)=0$ and that $E$ is the intersection of the subgraph of a function $u:K\rightarrow {\mathbb {R}}$ and the epigraph of a function $v:K\rightarrow {\mathbb {R}}$. The perimeter of $E$ is then given by

$$\begin{aligned} P_C(E)=\int _K\sqrt{1+|\nabla u|^2}\,dH^n+\int _K\sqrt{1+|\nabla v|^2}\,dH^n \ge 2H^n(K), \end{aligned}$$

with equality if and only if $\nabla u=\nabla v=0$. Hence $u, v$ are constant functions and $E$ is a slab. $\square$

As a consequence we have

Corollary 3.10

Let $K\subset {\mathbb {R}}^n$ be a convex body and $C= K\times [0,\infty )$. Then there is a constant $v_0>0$ such that any isoperimetric region in $M$ with volume $v\ge v_0$ is the slab $K\times [0,b]$, where $b=v/H^n(K)$. In particular, $I_C(v)=H^n(K)$ for $v\ge v_0$.

Proof

Just reflect with respect to the plane $x_{n+1}=0$ and apply Theorem 3.9. Alternatively, the proof of Theorem 3.9 can also be adapted to handle this case. $\square$

4 Cilindrically bounded convex sets

Given a cylindrically bounded convex body $C\subset {\mathbb {R}}^{n}\times {\mathbb {R}}$ so that $K$ is the closure of the orthogonal projection of $C$ over ${\mathbb {R}}^n\times \{0\}$, we shall say that $C_\infty =K\times {\mathbb {R}}$ is the asymptotic cylinder of $C$. Recall that, from our definition, $C$ is different from a cylinder. Assuming $C$ is unbounded in the positive vertical direction, the asymptotic cylinder can be obtained as a Hausdorff limit of downward translations of $C$. Another property of $C_\infty$ is the following: given $t\in {\mathbb {R}}$, define

$$\begin{aligned} C_t=C\cap ({\mathbb {R}}^n\times \{t\}). \end{aligned}$$

(4.1)

Then the orthogonal projection of $C_t$ to ${\mathbb {R}}^n\times \{0\}$ converges in Hausdorff distance to the basis $K$ of the asymptotic cylinder when $t\uparrow +\infty$ by [19, Thm. 1.8.16]. In particular, this implies

$$\begin{aligned} \lim _{t\rightarrow +\infty } H^n(C_t)=H^n(K). \end{aligned}$$

Let us prove now that the isoperimetric profile of $C$ is asymptotic to the one of the half-cylinder.

Theorem 4.1

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body with asymptotic cylinder $C_\infty =K\times {\mathbb {R}}$. Then

$$\begin{aligned} \lim _{v\rightarrow \infty } I_C(v)=H^n(K). \end{aligned}$$

(4.2)

Proof

We assume that $C$ is unbounded in the positive $x_{n+1}$-direction and consider the sets $\Omega (v)=C\cap ({\mathbb {R}}^n\times (-\infty ,t(v)])$, where $t(v)$ is chosen so that $|\Omega (v)|=v$. Then

$$\begin{aligned} I_C(v)\le P_C(\Omega (v))\le H^n(K), \end{aligned}$$

and taking limits we get

$$\begin{aligned} \limsup _{v\rightarrow \infty } I_C(v)\le H^n(K). \end{aligned}$$

Let us prove now that

$$\begin{aligned} H^n(K)\le \liminf _{v\rightarrow \infty } I_C(v). \end{aligned}$$

(4.3)

Fix $\varepsilon >0$. We consider a sequence of volumes $v_i\rightarrow \infty$ and a sequence $E_i\subset C$ of finite perimeter sets of volume $v_i$ with smooth boundary, so that

$$\begin{aligned} P_C(E_i)\le I_C(v_i) +\varepsilon . \end{aligned}$$

(4.4)

We shall consider two cases. Recall that $(E_i)_t=E_i\cap ({\mathbb {R}}^n\times \{t\})$.

Case 1. $\displaystyle \liminf _{i\rightarrow \infty }\left( \sup _{t>0}\ H^n((E_i)_t)\right) =H^n(K)$.

This is an easy case. Since the projection over the horizontal hyperplane does not increase perimeter we get

$$\begin{aligned} I_C(v_i)+\varepsilon \ge P_C(E_i)\ge \sup _{t>0}\ H^n((E_i)_t). \end{aligned}$$

Taking inferior limit, we get (4.3) since $\varepsilon >0$ is arbitrary.

Case 2. $\displaystyle \liminf _{i\rightarrow \infty }\left( \sup _{t>0}\ H^n((E_i)_t)\right) <H^n(K)$.

In this case, passing to a subsequence, there exists $v_0<H^n(K)$ such that $H^n((E_i)_t)\le v_0$ for all $t$. By [19, Thm. 1.8.16] we have $H^n(C_t)\rightarrow H^n(K)$. Hence there exists $t_0>0$ such that $v_0<H^n(C_t)$ for $t\ge t_0$. By Lemma 2.4, for $c_t=I_{C_t}(v_0)/v_0$, we get

$$\begin{aligned} I_{C_t}(v)\ge c_t v,\ \quad \text {for all}\ \quad v\le v_0,\ t\ge t_0. \end{aligned}$$

Furthermore, as $I_{C_t}(v_0) \rightarrow I_{K}(v_0)>0$ and $I_K(v_0)>0$, we obtain the existence of $c>0$ such that $c_t>c$ for $t$ large enough. Taking $t_0$ larger if necessary we may assume $c_t>c$ holds when $t\ge t_0$. Thus for large $i\in \mathbb {N}$ we obtain

$$\begin{aligned} |E_i|&=\int _0^{\infty }H^{n}((E_i)_t)\,dt\le b+\int _{t_0}^{\infty }H^{n}((E_i)_t)\,dt \\&\le b+\int _{t_0}^{\infty }c_t^{-1}H^{n-1}((\partial E_i)_t)\,dt \\&\le b+c^{-1}\int _{0}^{\infty }H^{n-1}((\partial E_i)_t)\,dt\le b+ c^{-1}P_C(E_i), \end{aligned}$$

where $b=t_0 H^{n}(K)$. So $P_C(E_i)\rightarrow \infty$ when $|E_i|\rightarrow \infty$. From (4.4) and $I_{C}\le H^{n}(K)$ we get a contradiction. This proves that Case 2 cannot hold and so (4.3) is proven. $\square$

Let us show now that the isoperimetric profile of $C$ is continuous and, when the boundary of $C$ is smooth enough, that the isoperimetric profile $I_C$ and its normalization $I_C^{(n+1)/n}$ are both concave non-decreasing functions. We shall need first some preliminary results.

Proposition 4.2

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex set, and $C_{\infty }=K\times {\mathbb {R}}$ its asymptotic cylinder. Consider a diverging sequence of finite perimeter sets $\{E_i\}_{i\in \mathbb {N}}\subset C$ such that $v=\lim _{i\rightarrow \infty }|E_i|$. Then

$$\begin{aligned} \liminf _{i\rightarrow \infty } P_C(E_i)\ge I_{C_{\infty }}(v) . \end{aligned}$$

Proof

Without loss of generality we assume $E_i\subset C\cap \{ x_{n+1}\ge i \}$. Let $r>0$ and $t_0>0$ so that the half-cylinder $B(0,r)\times [t_0,+\infty )$ is contained in $C\cap \{x_{n+1}\ge t_0\}$. Consider the horizontal sections $C_t=C\cap \{x_{n+1}=t\}, (C_\infty )_t=C_\infty \cap \{x_{n+1}=t\}$. We define a map $F:C\cap \{x_{n+1}\ge t_0\}\rightarrow C_\infty \cap \{x_{n+1}\ge t_0\}$ by

$$\begin{aligned} F(x,t)=(f_t(x),t), \end{aligned}$$

where $f_t:C_t\rightarrow (C_\infty )_t$ is defined as in (3.6) in [17]. For $i\in \mathbb {N}$, let $F_i=F|_{C\cap \{x_{n+1}\ge i\}}$. We will check that $\max \left\{ {{\mathrm{Lip}}}(F_i), {{\mathrm{Lip}}}\left( F_i^{-1}\right) \right\} \rightarrow 1$ when $i\rightarrow \infty$.

Take now $(x,t), (y,s)\in C\cap \{x_{n+1}\ge i\}$, and assume $t\ge s, i\ge t_0$. Then we have

$$\begin{aligned} |F(x,t)-F(y,s)|&=\left( |f_t(x)-f_s(y)|^2+|t-s|^2\right) ^{1/2} \nonumber \\&=\left( |f_t(x)-f_t(y)+f_t(y)-f_s(y)|^2+|t-s|^2\right) ^{1/2} \nonumber \\&=\left( |f_t(x)-f_t(y)|^2+|f_t(y)-f_s(y)|^2\right. \nonumber \\&\left. \quad +2\,|f_t(x)-f_t(y)||f_t(y)-f_s(y)|+|t-s|^2\right) ^{1/2} \end{aligned}$$

(4.5)

We have $|(f_t(x)-f_t(y))|\le {{\mathrm{Lip}}}(f_t)|x-y|$. By [17, Theorem 3.4], we can write ${{\mathrm{Lip}}}(f_t)<(1+\varepsilon _i)$ for $t\ge i$, where $\varepsilon _i\rightarrow 0$ when $i\rightarrow \infty$. Hence

$$\begin{aligned} |(f_t(x)-f_t(y))|\le (1+\varepsilon _i)\,|x-y|, \quad \text {for}\,\ t\ge i. \end{aligned}$$

(4.6)

We estimate now $|f_t(y)-f_s(y)|$. In case $|y|\le r$, we trivially have $|f_t(y)-f_s(y)|=0$. So we assume $|y|\ge r$. For $u\in \mathbb {S}^{n-1}$, consider the functions $\rho _t(u)=\rho (C_t,u), \rho (u)=\rho (K,u)$. Observe that, for every $u\in \mathbb {S}^{n}$ orthogonal to $\partial /\partial x_{n+1}$, the two-dimensional half-plane defined by $u$ and $\partial /\partial x_{n+1}$ intersected with $C$ is a two-dimensional convex set, and the function $t\mapsto \rho _t(u)$ is concave with a horizontal asymptotic line at height $\rho (u)$. So we have, taking $u=y/|y|$,

$$\begin{aligned} \frac{|f_t(y)-f_s(y)|}{|t-s|}=\frac{\left( |y|-r\right) }{|t-s|} \left| \frac{\rho _t(u)-r}{{\rho }(u)-r} -\frac{{\rho }_s(u)-r}{{\rho }(u)-r}\right| \le \frac{\left| {\rho }_t(u)-{\rho }_s(u)\right| }{|t-s|}, \end{aligned}$$

since $|y|-r\ge \rho (u)-r$. Using the concavity of $t\mapsto \rho _t(u)$ we get

$$\begin{aligned} \frac{\left| {\rho }_t(u)-{\rho }_s(u)\right| }{|t-s|}\le \left| \rho _i(u)-\rho _{i-1}(u)\right| , \quad \text {for}\,\ t,s\ge i. \end{aligned}$$

Letting $\ell _i=\sup _{u\in \mathbb {S}^{n-1}}|\rho _i(u)-\rho _{i-1}(u)|$, we get

$$\begin{aligned} |f_t(y)-f_s(y)|\le \ell _i\,|t-s|. \end{aligned}$$

(4.7)

As $C_{\infty }$ is the asymptotic cylinder of $C$ we conclude that $\ell _i\rightarrow 0$ when $i\rightarrow \infty$.

From (4.5) to (4.7), and trivial estimates, we obtain

$$\begin{aligned} |F_i(x,t)-F_i(y,s)|\le \left( (1+\varepsilon _i)^2+\ell _i^2+(1+\varepsilon _i)\,\ell _i\right) ^{1/2}\,|(x,t)-(y,s)| \end{aligned}$$

(4.8)

Now $\varepsilon _i\rightarrow 0$ and $\ell _i\rightarrow 0$ as $i\rightarrow \infty$. Thus inequality (4.8) yields

$$\begin{aligned} \limsup _{i\rightarrow \infty }{{\mathrm{Lip}}}(F_i)\le 1. \end{aligned}$$

Similarly we find $\limsup _{i\rightarrow \infty }{{\mathrm{Lip}}}\left( F_i^{-1}\right) \le 1$ and since ${{\mathrm{Lip}}}\left( F_i^{-1}\right) {{\mathrm{Lip}}}(F_i)\ge 1$ by Remark 2.2, we finally get $\max \left\{ {{\mathrm{Lip}}}(F_i),{{\mathrm{Lip}}}\left( F_i^{-1}\right) \right\} \rightarrow 1$ when $i\rightarrow \infty$.

Thus we have

$$\begin{aligned} v=\lim _{i\rightarrow \infty }|E_i|&=\lim _{i\rightarrow \infty } |F_i(E_i)|, \nonumber \\ \liminf _{i\rightarrow \infty } P_C(E_i)&= \liminf _{i\rightarrow \infty } P_{C_{\infty }}(F_i(E_i)). \end{aligned}$$

(4.9)

Now from (4.9) and the continuity of $I_{C_{\infty }}$ we get

$$\begin{aligned} \liminf _{i\rightarrow \infty } P_C(E_i)= \liminf _{i\rightarrow \infty } P_{C_{\infty }}(F_i(E_i))\ge I_{C_{\infty }}(v). \end{aligned}$$

$\square$

Lemma 4.3

Let $C \subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex set and $C_{\infty }=K\times {\mathbb {R}}$ its asymptotic cylinder. Let $E_\infty \subset C_{\infty }$ a bounded set of finite perimeter. Then there exists a sequence $\{E_i\}_{i\in \mathbb {N}}\subset C$ of finite perimeter sets such that $|E_i|=|E_\infty |$ and $\lim _{i\rightarrow \infty }P_C(E_i)=P_{C_{\infty }}(E_\infty )$.

Proof

Let $e_{n+1}=(0,\ldots ,0,1)\in {\mathbb {R}}^{n+1}$. We consider the truncated downward translations of $C$ defined by

$$\begin{aligned} C_i =(-i\,e_{n+1}+C)\cap \{t\ge 0\},\quad \ i\in \mathbb {N}. \end{aligned}$$

These convex bodies have the same asymptotic cylinder and

$$\begin{aligned} \bigcup _{i\in \mathbb {N}} C_i=C_{\infty }\cap [0,\infty ). \end{aligned}$$

(4.10)

Translating $E_\infty$ along the vertical direction if necessary we assume $E_\infty \subset \{t> 0\}$. Consider the sets $G_i=E_\infty \cap C_i$. For large indices $G_i$ is not empty by (4.10).

By the monotonicity of the Hausdorff measure we have $|G_i| \uparrow |E_\infty |$, and $H^{n}(\partial G_i\cap {{\mathrm{int}}}(C_i)) \uparrow H^{n}(\partial E_\infty \cap {{\mathrm{int}}}(C_\infty ))$. As $E_\infty$ is bounded, for large $i$ we can find Euclidean geodesic balls $B_i\subset {{\mathrm{int}}}(C_i)$, disjoint from $G_i$, such that $|B_i|=|E_\infty |-|G_i|$. Obviously the volume and and the perimeter of these balls go to zero when $i$ goes to infinity. Then $E_i=G_i\cup B_i$ are the desired sets. $\square$

Proposition 4.4

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body. Then $I_C$ is continuous.

Proof

Let $C_\infty =K\times {\mathbb {R}}$ be the asymptotic cylinder of $C$. The continuity of the isoperimetric profile $I_C$ at $v=0$ is proven by comparison with geodesic balls intersected with $C$.

Fix $v>0$ and let $\{v_i\}_{i\in \mathbb {N}}$ be a sequence of positive numbers converging to $v$. Let us prove first the lower semicontinuity of $I_C$. By the definition of isoperimetric profile, given $\varepsilon >0$, there is a finite perimeter set $E_i$ of volume $v_i$ so that $I_C(v_i)\le P_C(E_i)\le I_C(v_i)+\tfrac{1}{i}$, for every $i\in \mathbb {N}$. Reasoning as in [16, Thm. 2.1], we can decompose $E_i=E_i^c\cup E_i^d$ into convergent and diverging pieces, and there is a finite perimeter set $E\subset C$, eventually empty, so that

$$\begin{aligned} |E_i|&=|E_i^c|+|E_i^d|, \nonumber \\ P_C(E_i)&=P_C(E_i^c)+P_C(E_i^d), \nonumber \\ |E_i^c|&\rightarrow |E|, \nonumber \\ P_C(E)&\le \liminf _{i\rightarrow \infty } P_C(E_i^c). \end{aligned}$$

(4.11)

Let $w_1=|E|$. By Proposition 3.2, there exists an isoperimetric region $E_\infty \subset C_{\infty }$ of volume $|E_\infty |=w_2=v-w_1$. By Proposition 4.2 we have $P_{C_{\infty }}(E_\infty )\le \liminf _{i\rightarrow \infty } P_C(E_i^d)$. Hence

$$\begin{aligned} I_C(v)\le I_C(w_1)+I_{C_{\infty }}(w_2)&\le P_C(E)+P_{C_{\infty }}(E_\infty ) \\&\le \liminf _{i\rightarrow \infty } P_C(E_i^c)+\liminf _{i\rightarrow \infty } P_C(E_i^d) \\&\le \liminf _{i\rightarrow \infty } P_C(E_i) \\&=\liminf _{i\rightarrow \infty } I_C(v_i). \end{aligned}$$

To prove the upper semicontinuity of $I_C$ we shall use a variational argument. We can find bounded sets $E_0\subset C$, $E_\infty \subset C_\infty$ such that $|E_0|+|E_\infty |=v$ and $I_C(v)=P_C(E_0)+P_{C_\infty }(C_\infty )$. Deforming these sets and using Lemma 4.3, we easily show the existence of subsets $E_s\subset C$, with $|E_s|=v+s$, $s\in (-\delta ,\delta )$, such that

$$ |H^n(\partial _C E_s)-I_C(v)|\leqslant Ms, $$

for some uniform constant $M>0$. Hence

$$\begin{aligned} I_C(v+s)\leqslant H^n(\partial _C E_s)\leqslant I_C(v)+Ms. \end{aligned}$$

Taking a sequence $v_i\rightarrow v$ we get $\limsup _{i\rightarrow \infty }I_C(v_i)\leqslant I_C(v)$ and we obtain the upper semicontinuity of $I_C$. $\square$

Proposition 4.5

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body with asymptotic cylinder $C_\infty =K\times {\mathbb {R}}$. Assume that both $C$ and $C_\infty$ have smooth boundary. Then isoperimetric regions exist on $C$ for large volumes and have connected boundary. Moreover $I_C^{(n+1)/n}$ and so $I_C$ are concave non-decreasing functions.

Proof

Fix $v>0$. By [16, Thm. 2.1] there exists an isoperimetric region $E\subset C$ (eventually empty) of volume $|E|=v_1\le v$, and a diverging sequence $\{E_i\}_{i\in \mathbb {N}}$ of finite perimeter sets of volume $v_2=v-v_1$, such that

$$\begin{aligned} I_C(v)=P_C(E)+\lim _{i\rightarrow \infty }P_C(E_i) \end{aligned}$$

(4.12)

By Proposition 3.2, there is an isoperimetric region $E_\infty \subset C_\infty$ of volume $v_2$. We claim

$$\begin{aligned} \lim _{i\rightarrow \infty }P_C(E_i)= P_{C_{\infty }}(E_\infty ). \end{aligned}$$

(4.13)

If (4.13) does not hold, then Proposition 4.2 implies $\liminf _{i\rightarrow \infty } P_C(E_i)> I_{C_{\infty }}(v_2)$, and Lemma 4.3 provides a sequence of finite perimeter sets in $C$, of volume $v_2$, approaching $E_\infty$. This way we can build a minimizing sequence of sets of volume $v$ whose perimeters converge to some quantity strictly smaller than $I_C(v)$, a contradiction that proves (4.13). From (4.12) to (4.13) we get

$$\begin{aligned} I_C(v)=P_C(E)+P_{C_\infty }(E_\infty ). \end{aligned}$$

(4.14)

Reasoning as in the proof of Theorem 2.8 in [15], the configuration $E\cup E_\infty$ in the disjoint union of the sets $C, C_\infty$ must be stationary and stable, since otherwise we could slightly perturb $E\cup E_\infty$, keeping constant the total volume, to get a set ${E}^{\prime }\cup {E^{\prime }_\infty }$ such that

$$\begin{aligned} P_C(E^{\prime })+P_{C_{\infty }}(E^{\prime }_\infty )<P_C(E)+P_{C_{\infty }}({E_\infty }), \end{aligned}$$

contradicting (4.14).

Now as $C, C_{\infty }$ are convex and have smooth boundary, we can use a stability argument similar to that in [1, Proposition 3.9] to conclude that one of the sets $E$ or $E_\infty$ must be empty and the remaining one must have connected boundary. A third possibility, that $\partial _C E \cup \partial _{C_{\infty }} E_\infty$ consists of a finite number of hyperplanes intersecting orthogonally both $C$ and $C_{\infty }$, can be discarded since in this case $E_\infty$ would be a slab with $P_{C_\infty }(E_\infty )=2 H^n(K)>I_C$.

If $v$ is large enough so that isoperimetric regions in $C_\infty$ are slabs, then the above argument shows existence of isoperimetric regions of volume $v$ in $C$.

As $I_C$ is always realized by an isoperimetric set in $C$ or $C_{\infty }$, the arguments in [1, Theorem 3.2] imply that the second lower derivative of $I_C^{(n+1)/n}$ is non-negative. As $I_C^{(n+1)/n}$ is continuous by Proposition 4.4, Lemma 3.2 in [14] implies that $I_C^{(n+1)/n}$ is concave and hence non-decreasing. Then $I_C$ is also concave as a composition of $I_C^{(n+1)/n}$ with the concave non-increasing function $x\mapsto x^{n/(n+1)}$.

The connectedness of the isoperimetric regions in $C$ follows easily as an application of the concavity of $I_C^{(n+1)/n}$, as in [17, Theorem 4.6]. $\square$

The concavity of $I_C^{(n+1)/n}$ also implies the following Lemma. The proof in [17, Lemma 4.9] for convex bodies also holds in our setting.

Lemma 4.6

Let $C$ be be a cylindrically bounded convex body with asymptotic cylinder $C_\infty$. Assume that both $C$ and $C_\infty$ have smooth boundary. Let $\lambda \ge 1$. Then

$$\begin{aligned} I_{\lambda C}(v)\ge I_C(v) \end{aligned}$$

(4.15)

for all $0\le v\le |C|$.

Our aim now is to get a density estimate for isoperimetric regions of large volume in Theorem 4.9. This estimate would imply the convergence of the free boundaries of large isoperimetric regions to hyperplanes in Hausdorff distance given in Theorem 4.13.

Proposition 4.7

Let $C$ be cylindrically bounded convex body with asymptotic cylinder $C_{\infty }$. Given $r_0>0$, there exist positive constants $M, \ell _1$, only depending on $r_0$ and $C$, $C_{\infty }$, and a universal positive constant $\ell _2$ so that

$$\begin{aligned} P_{\overline{B}_C(x,r)}(v)\ge M\, {\min \{v,|\overline{B}_C(x,r)|-v\}}^{n/(n+1)}, \end{aligned}$$

(4.16)

for all $x\in C, 0<r\le r_0$, and $0<v<|\overline{B}(x,r)|$. Moreover

$$\begin{aligned} \ell _1 r^{n+1} \le |\overline{B}_C(x,r)| \le \ell _2 r^{n+1}, \end{aligned}$$

(4.17)

for any $x\in C, 0<r\le r_0$.

Proof

Reasoning as in [17, Theorem 4.12], it is enough to show

$$\begin{aligned} \Lambda _0=\inf _{x\in C}{{\mathrm{inr}}}(\overline{B}_C(x,r_0))>0. \end{aligned}$$

To see this consider a sequence $\{x_i\}_{i\in \mathbb {N}}$ so that ${{\mathrm{inr}}}(\overline{B}_C(x_i,r_0))$ converges to $\Lambda _0$. If $\{x_i\}_{i\in \mathbb {N}}$ contains a bounded subsequence then we can extract a convergent subsequence to some point $x_0\in C$ so that $\Lambda _0={{\mathrm{inr}}}(\overline{B}(x_0,r_0)>0$. If $\{x_i\}_{i\in \mathbb {N}}$ is unbounded, we translate vertically the balls $\overline{B}_C(x_i,r_0)$ so that the new centers $x_i^{\prime }$ lie in the hyperplane $x_{n+1}=0$. Passing to a subsequence we may assume that $x_i^{\prime }$ converges to some point $x_0\in C_\infty$. By the proof of Proposition 4.2, we have Hausdorff convergence of the translated balls to $\overline{B}_{C_\infty }(x_0,r_0)$ and so $\Lambda _0={{\mathrm{inr}}}(\overline{B}_{C_\infty }(x_0,r_0))>0$. $\square$

The next Lemma appeared in [17, Lemma 5.4]. We recall the proof here for completeness.

Lemma 4.8

For any $v>0$, consider the function $f_v:[0,v]\rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} f_v(s)=s^{-n/(n+1)}\,\bigg (\bigg (\frac{v-s}{v}\bigg )^{n/(n+1)}-1\bigg ). \end{aligned}$$

Then there is a constant $0<c_2<1$ that does not depends on $v$ so that $f_v(s)\ge -(1/2)\,v^{-n/(n+1)}$ for all $0\le s\le c_2\,v$.

Proof

By continuity, $f_v(0)=0$. Observe that $f_v(v)=-v^{-n/(n+1)}$ and that, for $s\in [0,1]$, we have $f_v(sv)=f_1(s)\,v^{-n/(n+1)}$. The derivative of $f_1$ in the interval $(0,1)$ is given by

$$\begin{aligned} f_1^{\prime }(s)=\frac{n}{n+1}\,\frac{(s-1)+(1-s)^{n/(n+1)}}{s-1}\,s^{-1-n/(n+1)}, \end{aligned}$$

which is strictly negative and so $f_1$ is strictly decreasing. Hence there exists $0<c_2<1$ such that $f_1(s)\ge -1/2$ for all $s\in [0,c_2]$. This implies $f_v(s)=f_1(s/v)\,v^{-n/(n+1)}\ge -(1/2)\,v^{-n/(n+1)}$ for all $s\in [0,c_2 v]$.

$\square$

Theorem 4.9

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body with asymptotic cylinder $C_\infty =K\times {\mathbb {R}}$. Assume that $C, C_\infty$ have smooth boundary. Let $E\subset C$ an isoperimetric region of volume $v>1$. Choose $\varepsilon$ so that

$$\begin{aligned} 0<\varepsilon <\bigg \{{\ell _2}^{-1},c_2,\frac{{\ell _2}^n}{8^{n+1}},{\ell _2}^{-1}\bigg (\frac{I_C(1)}{4}\bigg )^{n+1}\bigg \}, \end{aligned}$$

(4.18)

where $c_2$ is the constant in Lemma  4.8., and $\ell _1, \ell _2$ the constants in Proposition  4.7.

Then, for any $x\in C$ and $R\le 1$ so that $h(x,R)\le \varepsilon$, we get

$$\begin{aligned} h(x,R/2)=0. \end{aligned}$$

(4.19)

Moreover, in case $h(x,R)=|E\cap B_C(x,R)||B_C(x,R)|^{-1}$, we get $|E\cap B_C(x,R/2)|=0$ and, in case $h(x,R)=|B_C(x,R)\setminus E||B_C(x,R)|^{-1}$, we have $|B_C(x,R/2)\setminus E|=0$.

Proof

From the concavity of $I_C^{(n+1)/n}$ and the fact that $I_C(0)=0$ we get, as in Lemma 4.9 in [17], the following inequality

$$\begin{aligned} I_C(w)\ge c_1w^{n/(n+1)},\quad c_1=I_C(1), \end{aligned}$$

(4.20)

for all $0\le w\le 1$.

Assume first that

$$\begin{aligned} h(x,R)=\frac{|E\cap B_C(x,R)|}{|B_C(x,R)|}. \end{aligned}$$

Define $m(t)=|E\cap B_C(x,t)|, 0<t\le R$. Thus $m(t)$ is a non-decreasing function. For $t\le R\le 1$ we get

$$\begin{aligned} m(t)\le m(R)=|E\cap B_C(x,R)|=h(x,R) \,|B_C(x,R)|\le h(x,R)\,\ell _2 R^{n+1}\le \varepsilon \ell _2<1, \end{aligned}$$

(4.21)

by (4.18). Since $v>1$, we get $v-m(t)>0$.

By the coarea formula, when $m^{\prime }(t)$ exists, we obtain

$$\begin{aligned} m^{\prime }(t)=\frac{d}{dt}\int _0^tH^{n}(E\cap \partial _C B(x,s))\,ds=H^{n}(E\cap \partial _C B(x,t)). \end{aligned}$$

(4.22)

Define

$$\begin{aligned} \lambda (t)=\frac{v^{1/{(n+1)}}}{(v-m(t))^{1/{(n+1)}}},\quad E(t)=\lambda (t)(E\setminus B_C(x,t)). \end{aligned}$$

(4.23)

Then $E(t)\subset \lambda (t)C$ and $|E(t)|=|E|=v$. By Lemma 4.6, we get $I_{\lambda (t) C}\ge I_C$ since $\lambda (t)\ge 1$. Combining this with [23, Cor. 5.5.3], Equ. (4.22), and elementary properties of the perimeter functional, we have

$$\begin{aligned} I_C(v)&\le I_{\lambda (t) C}(v)\le P_{\lambda (t)C}(E(t))=\lambda ^n(t)\,P_C(E\setminus B_C(x,t)) \nonumber \\&\le \lambda ^n(t)\left( P_C(E)-P(E,B_C(x,t))+H^{n}(E\cap \partial B_C(x,t))\right) \nonumber \\&\le \lambda ^n(t)\left( P_C(E)-P_C(E\cap B_C(x,t))+2H^{n}(E\cap \partial B_C(x,t))\right) \nonumber \\&\le \lambda ^n(t)\left( I_C(v)-c_1m(t)^{n/{(n+1)}}+2m^{\prime }(t)\right) , \end{aligned}$$

(4.24)

where $c_1$ is the constant in (4.20). Multiplying both sides by $I_C(v)^{-1}{\lambda (t)}^{-n}$ we find

$$\begin{aligned} {\lambda (t)}^{-n}-1+\frac{c_1}{I_C(v)}m(t)^{n/{(n+1)}}\le \frac{2}{I_C(v)}m^{\prime }(t). \end{aligned}$$

(4.25)

As we have $I_C\le H^n(K)$, and $I_C$ is concave by Proposition 4.5, there exists a constant $\alpha >0$ such that $I_C\ge \alpha$ for sufficient large volumes. Set

$$\begin{aligned} a=\frac{2}{\alpha }\ge \frac{2}{I_C(v)},\quad \text {and}\quad b=\frac{c_1}{H^n(K)}\le \frac{c_1}{I_C(v)}. \end{aligned}$$

(4.26)

From the definition (4.23) of $\lambda (t)$ we get

$$\begin{aligned} f(m(t))\le am^{\prime }(t)\,\quad H^1\text {-a.e}, \end{aligned}$$

(4.27)

where

$$\begin{aligned} \frac{f(s)}{s^{n/(n+1)}}=b+\frac{\left( \frac{v-s}{v}\right) ^{n/{(n+1)}}-1}{s^{n/(n+1)}}. \end{aligned}$$

(4.28)

By Lemma 4.8, there exists a universal constant $0<c_2<1$, not depending on $v$, so that

$$\begin{aligned} \frac{f(s)}{s^{n/{n+1}}}\ge b/2\quad \text {whenever}\quad 0< s\le c_2 . \end{aligned}$$

(4.29)

Since $\varepsilon \le c_2$ by (4.18), Eq. (4.29) holds in the interval $[0,\varepsilon ]$. If there were $t\in [R/2,R]$ such that $m(t)=0$ then, by monotonicity of $m(t)$, we would conclude $m(R/2)=0$ as well. So we assume $m(t)>0$ in $[R/2,R]$. Then by (4.27) and (4.29), we get

$$\begin{aligned} b/2a\le \frac{m^{\prime }(t)}{m(t)^{n/{n+1}}}, \quad H^1\text {-a.e.} \end{aligned}$$

Integrating between $R/2$ and $R$ we get by (4.21)

$$\begin{aligned} bR/4a\le \left( m(R)^{1/{(n+1)}}-m(R/2)^{1/{(n+1)}}\right) \le m(R)^{1/{(n+1)}}\le (\varepsilon \ell _2)^{1/(n+1)}R. \end{aligned}$$

This is a contradiction, since $\varepsilon \ell _2<(b/4a)^{n+1}=I_C(v)^{n+1}/\left( 8^{n+1} v^n\right) \le \ell _2^{n+1}/8^{n+1}$ by (4.18) and Proposition 2.6. So the proof in case $h(x,R)=|E\cap B_C(x,R)|\,(|B_C(x,R))|^{-1}$ is completed. For the remaining case, when $h(x,R)=|B_C(x,R)|^{-1}|B_C(x,R)\setminus E|$, we use Lemma 2.4 and the fact that $I_C$ is non-decreasing proven in Proposition 4.5. Then we argue as in Case 1 in Lemma 4.2 of [8] to get

$$\begin{aligned} c_1/4\le (\varepsilon \ell _2)^{1/(n+1)}. \end{aligned}$$

This is a contradiction, since $\varepsilon \ell _2<(c_1/4)^{n+1}$ by assumption (4.18) $\square$

Proposition 4.10

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body and $C_\infty$ its asymptotic cylinder. Assume that both $C$ and $C_\infty$ have smooth boundary. Then there exists a constant $c>0$ such that, for each isoperimetric region $E$ of volume $v>1$,

$$\begin{aligned} P(E,B_C(x,r))\ge cr^n, \end{aligned}$$

(4.30)

for $r\le 1$ and $x\in \partial _C E$.

Proof

Let $E\subset C$ be an isoperimetric region of volume larger than $1$. Choose $\varepsilon >0$ satisfying (4.18). Since $x\in \partial _C E$ we have $\lim _{r\rightarrow 0} h(x,r)\ne 0$ and, by Theorem 4.9, $h(x,r)\ge \varepsilon$ for $0<r\le 1$. So we get

$$\begin{aligned} P(E,B_C(x,r))&\ge M\min \left\{ |E\cap B_C(x,r)|, |B_C(x,r) \setminus E|\right\} ^{n/{(n+1)}} \\&=M\,(|B_C(x,r)|\,h(x,r))^{n/(n+1)}\ge M(|B_C(x,r)|\,\varepsilon )^{n/(n+1)} \\&\ge M\,(\ell _1\varepsilon )^{n/(n+1)}\,r^n. \end{aligned}$$

Inequality (4.30) follows by taking $c=M(\ell _1\varepsilon )^{n/(n+1)}$, which is independent of $v$. $\square$

Remark 4.11

Theorem 4.9 and Proposition 4.10 also hold if $C$ is a convex cylinder.

As a Corollary we obtain a new proof of Theorem 3.9

Corollary 4.12

Let $C=K\times {\mathbb {R}}$, where $K\subset {\mathbb {R}}^n$ is a convex body. Then there is a constant $v_0>0$ so that $I_C(v)=2H^n(K)$ for all $v\ge v_0$. Moreover, the slabs $K\times [t_1,t_2]$ are the only isoperimetric regions of volume larger than or equal to $v_0$.

Proof

Let $E$ be an isoperimetric region with volume

$$\begin{aligned} |E|>2mr_0 H^n(K), \end{aligned}$$

(4.31)

where $r_0, c>0$, are the constants in Proposition 4.10 (see also Remark 4.11), and $m>0$ is chosen so that

$$\begin{aligned} mcr_0^n>2H^n(K). \end{aligned}$$

(4.32)

By results of Talenti on Steiner symmetrization for finite perimeter sets [22], we can assume that the boundary of $E$ is the union of two graphs, symmetric with respect to a horizontal hyperplane, over a subset $K^*\subset K$. If $K^*=K$ then $P_C(E)\ge 2 H^n(K)$, since the orthogonal projection over $K\times \{0\}$ is perimeter non-increasing. This implies $P_C(E)= 2 H^n(K)$ and it follows, as in the proof of Theorem 3.9, that $E$ is a slab.

So assume that $K^*$ is a proper subset of $K$. Since $|E|>2mr_0 H^n(K), E$ cannot be contained in the slab $K \times [-r_0m, r_0m]$. Then as $\partial _C E$ is a union of two graphs over $K^*$ we can find $x_j\in \partial _{C} E, 1\le j \le m$, so that the balls centered at these points are disjoint. Then by the lower density bound (4.30) we get

$$\begin{aligned} P_C(E)\ge \sum _{j=1}^m P(E,B_C(x_j,r_0))\ge mcr_0^n>2H^n(K), \end{aligned}$$

(4.33)

a contradiction since $I_C\le 2H^n(K)$. $\square$

Recall that, in Corollary 3.10, we showed that, given a half-cylinder $K\times [0,\infty )$, there exists $v_0>0$ so that every isoperimetric region in $K\times [0,\infty )$ of volume larger than or equal to $v_0$ is a slab $K\times [0,b]$, where $b=v/H^n(K)$. We can use this result to obtain

Theorem 4.13

Let $C\subset {\mathbb {R}}^{n+1}$ be a cylindrically bounded convex body, $C_{\infty }=K\times {\mathbb {R}}$ its asymptotic cylinder and $C_\infty ^+=K\times [0,\infty )$. Let $\{E_i\}_{i\in \mathbb {N}}$ be a sequence of isoperimetric regions with $\lim _{i\rightarrow \infty }|E_i|=\infty$.

Then truncated downward translations of $E_i$ converge in Hausdorff distance to a half-slab $K\times [0,b]$ in $C_\infty ^+$. The same convergence result holds for their free boundaries.

Proof

By Corollary 3.10, we can choose $v_0>0$ such that each isoperimetric region with volume $v\ge v_0$ in $C_+^{\infty }$ is a half-slab $K\times [0,b(v)]$ of perimeter $H^n(K)$, where $b(v)=v/H^n(K)$.

Since $|E_i|\rightarrow \infty$, we can find vertical vectors $y_i$, with $|y_i|\rightarrow \infty$, so that $\Omega _i=(-y_i+E_i)\cap \{x_{n+1}\ge 0\}$ has volume $v_0$ for large enough $i\in \mathbb {N}$. We observe also that, by Proposition 4.10 and the fact that $I_C\le H^n(K)$, the sets $\partial E_i$ have uniformly bounded diameter.

Consider the convex bodies

$$\begin{aligned} C_i=(-y_i+ C )\cap \{x_{n+1}\ge 0\}, \end{aligned}$$

(4.34)

for $i\in \mathbb {N}$. The sets $C_i$ have the same asymptotic cylinder $C_\infty$ and we have

$$\begin{aligned} \bigcup _{i\in \mathbb {N}} C_i=C^+_{\infty }. \end{aligned}$$

(4.35)

By construction

$$\begin{aligned} P_{C_i}(\Omega _i)\le P_C(E_i)\le H^n(K). \end{aligned}$$

(4.36)

Since $\partial E_i$ are uniformly bounded and $|\Omega _i|=v_0$, there exists a Euclidean geodesic ball $B$ such that $\Omega _i\subset B$ for all $i\in \mathbb {N}$. By (4.35) the sequence of convex bodies $\{C_i\cap B\}_{i\in \mathbb {N}}$ converges to $C_\infty ^+\cap B$ in Hausdorff distance and, by [17, Theorem 3.4], in lipschitz distance.

Hence, by the proof of [17, Theorem 3.4] and [17, Lemma 2.3], we conclude there exists a finite perimeter set $\Omega \subset C_\infty ^+$, such that

$$\begin{aligned} \Omega _i\mathop {\rightarrow }\limits ^{L^1}\Omega \quad \text {and}\quad P_{C_\infty ^+}(\Omega )\le \liminf _{i\rightarrow \infty } P_{C_i}(\Omega _i). \end{aligned}$$

(4.37)

So we obtain from (4.36) and (4.37),

$$\begin{aligned} H^n(K)=I_{C_\infty ^+}(v_0)\le P_{C_\infty ^+}(\Omega )\le \liminf _{i\rightarrow \infty } P_{C_i}(\Omega _i) \le \liminf _{i\rightarrow \infty } P_C(E_i)\le H^n(K), \end{aligned}$$

(4.38)

what implies that $\Omega$ is an isoperimetric region of volume $v_0$ in $C_\infty ^+$ and so it is a slab.

Furthermore, the arguments of [17, Theorem 5.11] and [17, Theorem 5.13] can be applied here to improve the $L^1$ convergence to Hausdorff convergence, both for the sets $\Omega _i$ and for their free boundaries.$\square$

Remark 4.14

The proof of Theorem 4.13 implies $\displaystyle \lim _{v\rightarrow \infty }I_C(v)=H^n(K)$. This way we get a different proof of Theorem 4.1.