The $\phi$-Brunn–Minkowski inequalities for general convex bodies

Abstract

In this paper, we first give a new proof of the log-Minkowski inequality of general planar convex bodies and then extend the $L_p$-Brunn–Minkowski inequality and $L_p$-Minkowski inequality of o-symmetric planar convex bodies for $p\in (0,1)$ to $\phi$-Brunn–Minkowski inequality and $\phi$-Minkowski inequality of general planar convex bodies. As an application, a family of $\phi$-measures of asymmetry for planar convex bodies is introduced.

1 Introduction

The classical Brunn–Minkowski inequality for convex bodies (compact convex sets with nonempty interiors) states that for convex bodies K, L in Euclidean n-space, $\mathbb {R}^n$, the volume of the bodies and of their Minkowski sum $K+L=\{x+y: x\in \text {and}\; y\in L\}$, are related by

$$\begin{aligned} V\left(K+ L\right )^{\frac{1}{n}} \ge V(K)^{\frac{1}{n}}+ V(L)^{\frac{1}{n}}, \end{aligned}$$

(1)

with equality if and only if K and L are homothetic.

The Brunn–Minkowski inequality has an equivalent formulation as for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )K+ \lambda L\right )\ge V(K)^{1-\lambda }V(L)^{\lambda }, \end{aligned}$$

(2)

and for $\lambda \in (0,1)$, there is equality if and only if K and L are translates.

The excellent survey article of Gardner [3] gives a comprehensive account of various aspects and consequences of the Brunn–Minkowski inequality.

In the 1960s, Firey [2] introduced for $p\ge 1$ the so-called Minkowski–Firey $L_p$ sum of convex bodies that contain the origin in their interiors, and established the $L_p$-Brunn–Minkowski inequality, which states as follows:

$$\begin{aligned} V\left((1-\lambda )\cdot K+_p\lambda \cdot L\right )^{\frac{p}{n}} \ge (1-\lambda )V(K)^{\frac{p}{n}}+\lambda V(L)^{\frac{p}{n}}, \end{aligned}$$

(3)

with equality for $\lambda \in (0,1)$ if and only if K and L are dilates.

In the mid-1990s, it was shown in Refs. [12, 13] that a study of the volume of $L_p$-Minkowski addition leads to an $L_p$-Brunn–Minkowski theory. This theory has expanded rapidly.

If K and L are convex bodies that contain the origin in their interiors and $0\le \lambda \le 1$ then the Minkowski–Firey $L_p$-combination ($p>0$), $(1-\lambda )\cdot K+_p\lambda \cdot L$, is defined by

$$\begin{aligned} (1-\lambda )\cdot K+_p\lambda \cdot L=\bigcap _{u\in S^{n-1}}\{x\in \mathbb {R}^n: x\cdot u\le \left((1-\lambda )h_K(u)^p+\lambda h_L(u)^p\right )^{1/p}\}. \end{aligned}$$

(4)

It has been noticed that the $L_p$-Minkowski addition makes sense for all $p>0$. The case $p=0$ is known as the log-Minkowski addition, $(1-\lambda )\cdot K+_0 \lambda \cdot L$, of convex bodies K and L that contain the origin in their interior, defined by

$$\begin{aligned} (1-\lambda )\cdot K+_0\lambda \cdot L= \bigcap _{u\in S^{n-1}}\{x\in \mathbb {R}^n: x\cdot u \le h_K(u)^{1-\lambda }h_L(u)^{\lambda }\}. \end{aligned}$$

(5)

In Ref. [1], Böröczky, Lutwak, Yang and Zhang conjectured the log-Brunn–Minkowski inequality: If K and L are o-symmetric convex bodies in $\mathbb {R}^n$, then for all $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_0\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }. \end{aligned}$$

(6)

The log-Brunn–Minkowski inequality is stronger than the $L_p$-Brunn–Minkowski inequality for $p>0$. It was shown in Ref. [1] that the log-Brunn–Minkowski inequality is equivalent to the following log-Minkowski mixed volume inequality: If K and L are o-symmetric convex bodies in $\mathbb {R}^n$, then

$$\begin{aligned} \int _{S^{n-1}}\log \frac{h_L}{h_K}d\bar{V}_K\ge \frac{1}{n} \log \frac{V(L)}{V(K)}. \end{aligned}$$

(7)

Here $\bar{V}_K$ denotes the cone-volume probability measure of K.

The planar case of the log-Brunn–Minkowski inequality and the equivalent log-Brunn–Minkowski inequality were proved in Ref. [1].

Theorem 1.1

([1]) If K and L are o-symmetric convex bodies in $\mathbb {R}^2$, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_0\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(8)

with equality for $\lambda \in (0,1)$ if and only if K and L are dilates or K and L are parallelograms with parallel sides.

Theorem 1.2

([1]) If K and L are o-symmetric convex bodies in $\mathbb {R}^2$, then,

$$\begin{aligned} \int _{S^1}\log \frac{h_L}{h_K}d\bar{V}_K\ge \frac{1}{2} \log \frac{V(L)}{V(K)}, \end{aligned}$$

(9)

with equality if and only if K and L are dilates or K and L are parallelograms with parallel sides.

It is easily seen from definition (4) that for fixed convex bodies K, L and fixed $\lambda \in [0,1]$, the $L_p$-Minkowski–Firey combination $(1-\lambda )\cdot K+_p\lambda \cdot L$ is increasing with respect to set inclusion, as p increases, i.e., if $0\le p\le q$,

$$\begin{aligned} (1-\lambda )\cdot K+_p\lambda \cdot L\subset (1-\lambda )\cdot K+_q\lambda \cdot L. \end{aligned}$$

(10)

From (9), the $L_p$-Brunn–Minkowski inequality and the $L_p$-Minkowski inequality were proved in Ref. [1] for $p\in (0,1)$.

Theorem 1.3

([1]) Suppose $0<p<1$. If K and L are o-symmetric convex bodies in $\mathbb {R}^2$, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_p\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(11)

with equality for $\lambda \in (0,1)$ if and only if $K=L$.

Theorem 1.4

([1]) Suppose $0<p<1$. If K and L are o-symmetric convex bodies in $\mathbb {R}^2$, then for all $\lambda \in [0,1]$,

$$\begin{aligned} \left( \int _{S^1}\left( \frac{h_L}{h_K}\right) ^pd\bar{V}_K\right) ^{\frac{1}{p}}\ge \left( \frac{V(L)}{V(K)}\right) ^{\frac{1}{2}}, \end{aligned}$$

(12)

with equality for $\lambda \in (0,1)$ if and only if K and L are dilates.

In Ref. [18], Ma gave an alternative proof of Theorem 1.2. Some results of the log-Brunn–Minkowski inequality for $n\ge 3,$ see Refs. [19, 21, 25].

There is a counterexample, showing that, if K is an o-centered cube and L is a distinct translate of K, then (6) does not hold for general non-o-symmetric convex bodies. By introducing the notion of “dilation position”, Xi and Leng [23] proved the log-Brunn–Minkowski inequality and the equivalent log-Minkowski mixed volume inequality for general planar convex bodies.

Theorem 1.5

([23]) If K and L are convex bodies in $\mathbb {R}^2$ with $o\in K\cap L$, and K, L are in dilation position, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_0\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(13)

with equality for $\lambda \in (0,1)$ if and only if K and L are dilates or K and L are parallelograms with parallel sides.

Theorem 1.6

([24]) If K and L are convex bodies in $\mathbb {R}^2$ with $o\in K\cap L$, and K, L are in dilation position, then

$$\begin{aligned} \int _{S^1}\log \frac{h_L}{h_K}d\bar{V}_K\ge \frac{1}{2} \log \frac{V(L)}{V(K)}, \end{aligned}$$

(14)

with equality if and only if K and L are dilates or K and L are parallelograms with parallel sides.

The Orlicz–Brunn–Minkowski theory originated with the work of Lutwak et al. [15, 16]. By introducing the Orlicz–Minkowski addition, Gardner, Hug and Weil [4], and Xi et al. [24] proved the Orlicz–Brunn–Minkowski inequality and Orlicz–Minkowski inequality. It is a natural extension of the $L_p$-Brunn–Minkowski theory for $p\ge 1$. For dual Orlicz–Brunn–Minkowski theory see [5, 26].

Let $\Phi$ be the set of strictly increasing functions $\phi : (0,\infty )\rightarrow I\subset \mathbb {R}$ which are continuously differentiable on $(0,\infty )$ with positive derivative, and satisfy that $\lim _{t\rightarrow \infty }\phi (t)=\infty$ and that $\log \circ \phi ^{-1}$ is concave. Observe that whenever $\phi \in \Phi$ is convex, the composite function $\log \circ \phi ^{-1}$ is concave. The collection of convex functions from $\Phi$ shall be denoted by $\mathcal {C}$.

Let $\lambda \in [0,1]$ and $\phi \in \Phi$. For $u\in S^{n-1}$, we define a function $h_{\lambda }(u)$ as

$$\begin{aligned} h_{\lambda }(u)=\inf \{\tau >0: (1-\lambda )\phi \left(\frac{h_K(u)}{\tau }\right)+\lambda \phi \left(\frac{h_L(u)}{\tau }\right)\le \phi (1)\}. \end{aligned}$$

(15)

By the strict monotonicity of $\phi$, we have

$$\begin{aligned} \phi (1)=(1-\lambda )\phi \left(\frac{h_K(u)}{h_{\lambda }(u)}\right)+\lambda \phi \left(\frac{h_L(u)}{h_{\lambda }(u)}\right). \end{aligned}$$

(16)

The $\phi$-combination $(1-\lambda )\cdot K+_{\phi } \lambda \cdot L$ of $K, L\in \mathcal {K}^n_o$ is defined in Ref. [17] by

$$\begin{aligned} (1-\lambda )\cdot K+_{\phi } \lambda \cdot L=\bigcap _{u\in S^{n-1}}\{x\in \mathbb {R}^n: x\cdot u\le h_{\lambda }(u)\}. \end{aligned}$$

(17)

Note that if $\phi (t)=t^p$ with $p>0$, then the $\phi$-combination reduces to the $L_p$-Minkowski combination. Further, if $\phi (t)=\alpha \log (t) (\alpha >0)$, then we retrieve the log-Minkowski combination. In Ref. [17], Lv proved the $\phi$-Minkowski inequality and $\phi$-Brunn–Minkowski inequality for general functions $\phi$ for o-symmetric planar convex bodies K, L. If $\phi (t)=t^p, p\in (0,1)$, then the $\phi$-Minkowski inequality reduces to the $L_p$-Minkowski inequality (12) and $L_p$-Brunn–Minkowski inequality (11).

In this paper, we first present a new proof Theorem 1.6, and extend Theorems 1.3 and 1.4 from $p\in (0,1)$ and o-symmetric convex bodies K, Lto general case $\phi$ and general convex bodies K, L. More precisely, we have the following main results.

Theorem 1.7

Let $\phi \in \Phi$ with $\phi \ne \alpha \log (\alpha >0)$, and K and L are planar convex bodies containing the origin o in their interiors, and $o\in K\cap L$. If K and L are at a dilation position, then

$$\begin{aligned} \int _{S^1}\phi \left( \frac{h_L}{h_K}\right) d\bar{V}_K\ge \phi \left( \frac{V(L)^{\frac{1}{2}}}{V(K)^{\frac{1}{2}}}\right), \end{aligned}$$

(18)

with equality if and only if K and L are dilates.

Theorem 1.8

Let $\phi \in \Phi$, $\phi \ne \alpha \log (\alpha >0)$ be concave on $(0,\infty )$, and K and L are planar convex bodies containing the origin o in their interiors, and $o\in K\cap L$. If K and L are at a dilation position, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_{\phi }\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(19)

with equality for $\lambda \in (0,1)$ if and only if $K=L$.

2 Preliminaries

Let $\mathcal {K}^n$ be the class of convex bodies (compact convex sets with nonempty interiors) in $\mathbb {R}^n$, and let $\mathcal {K}^n_o$ be those sets in $\mathcal {K}^n$ containing the origin in their interiors.

The support function $h_K: \mathbb {R}^n\rightarrow \mathbb {R}$, of compact convex subset K of $\mathbb {R}^n$ is defined by $h_K(x)=\{x\cdot y: y\in K\}$, for $x\in \mathbb {R}^n$, and uniquely determines the convex set.

A boundary point $x\in \partial K$ of the convex body K is said to have $u\in S^{n-1}$ as one of its outer unit normals provided $x\cdot u=h_K(u)$. A boundary point is said to be singular if it has more than one unit normal vector. It is well known that the set of singular boundary points of a convex body has $(n-1)$-dimensional Hausdorff measure $\mathcal {H}^{n-1}$ equal to 0.

Let $K\in \mathcal {K}^n$ and $\nu _K:\partial K\rightarrow S^{n-1}$ the generalized Gauss map. For each Borel set $\omega \subset S^{n-1}$, the inverse spherical image $\nu _K^{-1}(\omega )$ of $\omega$ is the set of all boundary points of K which have an outer unit normal belonging to the set $\omega$. The surface area measure $S_K$ of $K\in \mathcal {K}^n$ is defined by

$$\begin{aligned} S_K(\omega )=\mathcal {H}^{n-1}(\nu _K^{-1}(\omega )), \end{aligned}$$

(20)

for each Borel set $\omega \subset S^{n-1}$, i.e., $S_K(\omega )$ is the $(n-1)$-dimensional Hausdorff measure of the set of all points on $\partial K$ that have a unit normal that lies in $\omega$.

The Hausdorff distance $d_H(K,L)$ of compact convex sets K, L is defined by $d_H(K,L)=\Vert h_K-h_L\Vert _{\infty }$. A sequence of convex bodies, $K_i$, is said to converge to a body K, i.e., $\lim _{i\rightarrow \infty }K_i=K$ if $d_H(K_i,K)\rightarrow 0$. If K is a convex body and $K_i$ is a sequence of convex bodies then

$$\begin{aligned} \lim _{i\rightarrow \infty }K_i=K \Rightarrow \lim _{i\rightarrow \infty }S_{K_i}=S_K, \;\text {weakly}. \end{aligned}$$

(21)

The cone-volume measure $V_K$ of $K\in \mathcal {K}^n$ is a Borel measure on the unit sphere $S^{n-1}$ defined for a Borel set $\omega \subset S^{n-1}$ by

$$\begin{aligned} V_K(\omega )=\frac{1}{n} \int _{x\in \nu _K^{-1}(\omega )}x\cdot \nu _K(x) \text{d}\mathcal {H}^{n-1}(x), \end{aligned}$$

(22)

and thus

$$\begin{aligned} \text{d}V_K=\frac{1}{n} h_KdS_K. \end{aligned}$$

(23)

Since,

$$\begin{aligned} V(K)=\frac{1}{n}\int _{u\in S^{n-1}}h_K(u)\text{d}S_K(u), \end{aligned}$$

(24)

we can define the cone-volume probability measure $\bar{V}_K$ of K by

$$\begin{aligned} \bar{V}_K=\frac{1}{V(K)}V_K. \end{aligned}$$

(25)

Suppose $K,L\in \mathcal {K}^n_o$. For $p\ne 0$, the $L_p$-mixed volume $V_p(K,L)$ can be defined as

$$\begin{aligned} V_p(K,L)=\int _{u\in S^{n-1}}\left( \frac{h_L}{h_K}\right) ^p\text{d}V_K. \end{aligned}$$

(26)

The normalized $L_p$-mixed volume $\bar{V}_p(K,L)$ was first defined in Ref. [14],

$$\begin{aligned} \bar{V}_p(K,L)=\left(\int _{u\in S^{n-1}}\left( \frac{h_L}{h_K}\right) ^p\text{d}\bar{V}_K\right )^{\frac{1}{p}}. \end{aligned}$$

(27)

For $p=\infty$, we define

$$\begin{aligned} \bar{V}_{\infty }(K,L)=\max \{h_L/h_K: u\in \text {supp}S_K \}, \end{aligned}$$

(28)

and we have

$$\begin{aligned} \lim _{p\rightarrow \infty } \bar{V}_p(K,L)=\bar{V}_{\infty }(K,L). \end{aligned}$$

(29)

Letting $p\rightarrow 0$ gives

$$\begin{aligned} \bar{V}_0(K,L)=\exp \left( \int _{u\in S^{n-1}}\log \frac{h_L}{h_K}\text{d}\bar{V}_K\right) , \end{aligned}$$

(30)

which is the normalized log-mixed volume of K and L. From Jesen’s inequality we know that $p\mapsto \bar{V}_p(K,L)$ is strictly monotone increasing, unless $h_L/h_K$ is constant on $\text {supp}S_K$.

Suppose $K,L\in \mathcal {K}^n$. The inradius r(K, L) and R(K, L) of K with respect to L are defined by

$$\begin{aligned} r(K,L)=\sup \{t>0: x+tL \subset K \;\text {and}\; x\in \mathbb R^n\}, \end{aligned}$$

$$\begin{aligned} R(K,L)=\inf \{t>0: x+tL \supset K \;\text {and}\; x\in \mathbb R^n\}. \end{aligned}$$

From the definition, it follows that $r(K,L)=1/R(L,K)$.

If K, L happen to be o-symmetric convex bodies, then clearly

$$\begin{aligned} r(K,L)=\min _{u\in S^{n-1}}\frac{h_K(u)}{h_L(u)} \;\text {and}\; R(K,L)=\max _{u\in S^{n-1}}\frac{h_K(u)}{h_L(u)}. \end{aligned}$$

(31)

Let $K,L \in \mathcal K^n$. K and L are said to be at a dilation position, if $o\in K\cap L$, and

$$\begin{aligned} r(K,L)L \subset K \subset R(K,L) L. \end{aligned}$$

(32)

The definition and some properties of dilation position were first given by Xi and Leng [23]. It is easy to prove that if K, L are o-symmetric convex bodies, then K and L are at a dilation position.

In general, we refer the reader to [20] for standard notation concerning convex bodies.

3 A new proof of Theorem 1.6

In Ref. [18], Ma gave a proof of Theorem 1.1. In the following, we demonstrate an alternate proof of Theorem 1.5 by employing Ma’s approach [18]. The following lemma is needed in our proof.

Lemma 3.1

([23]) Let $K,L\in \mathcal K^2$ with $o\in K\cap L$. If K and L are at a dilation position, then

$$\begin{aligned} \int _{S^1}\frac{h_K}{h_L}\text{d}\bar{V}_K \le \frac{V(L,K)}{V(L)}, \end{aligned}$$

(33)

with equality if and only if K and L are dilates, or K and L are parallelograms with parallel sides.

We repeat the statement of Theorem 1.6, and present our approach.

Theorem 3.2

([23]) If K and L are convex bodies in $\mathbb {R}^2$ with $o\in K\cap L$, and K, L are at a dilation position, then

$$\begin{aligned} \int _{S^1}\log \frac{h_L}{h_K}\text{d}\bar{V}_K\ge \frac{1}{2} \log \frac{V(L)}{V(K)}, \end{aligned}$$

(34)

with equality if and only if K and L are dilates or K and L are parallelograms with parallel sides.

Proof

Set

$$\begin{aligned} F(t)= \int _{S^1}\log \left( \frac{h_{L+tK}}{h_K}\right) \text{d}\bar{V}_K-{\frac{1}{2}}\log \left( \frac{V(L+tK)}{V(K)}\right) , \;\;t\in [0,\infty ). \end{aligned}$$

(35)

Since $h_{L+tK}=h_L+th_K$ and $V(L+tK)=V(L)+2V(L,K)t+V(K)t^2$, we have

$$\begin{aligned} F'(t)= & {} \int _{S^1}\frac{h_K}{h_L+th_K}\text{d}\bar{V}_K-\frac{(V(L,K)+V(K)t)}{V(L)+2V(L,K)t+V(K)t^2}\\= & {} \int _{S^1}\frac{h_K}{h_{L+tK}}\text{d}\bar{V}_K-\frac{V(L+tK,K)}{V(L+tK)}. \end{aligned}$$

By Lemma 5.2 of Ref. [23], we have K and $L+tK$ are at a dilation position. Therefore, we get $F'(t)\le 0$ from Lemma3.1, which implies that F(t) is decreasing on $[0,\infty )$.

By mean value theorem for integrals, there exists $u_0\in S^1$ such that

$$\begin{aligned} \int _{S^1}\log \left(\frac{h_{L+tK}}{h_K}\right )\text{d}\bar{V}_K=\log \left(\frac{h_{L+tK}(u_0)}{h_K(u_0)}\right ). \end{aligned}$$

(36)

Let $t\rightarrow \infty$, then

$$\begin{aligned} F(t)= & {} \log \left( \frac{h_{L+tK}(u_0)}{h_K(u_0)}\right) -{\frac{1}{2}}\log \left (\frac{V(L+tK)}{V(K)}\right )\\= & {} \log \left(\frac{h_L(u_0)+th_K(u_0)}{h_K(u_0)}\cdot \frac{V(K)^{\frac{1}{2}}}{V(L+tK)^{\frac{1}{2}}}\right )\\= & {} \log \left(\frac{h_L(u_0)+th_K(u_0)}{h_K(u_0)}\cdot \frac{V(K)^{\frac{1}{2}}}{(V(L)+2tV(L,K)+t^2V(K))^{\frac{1}{2}}}\right )\\\rightarrow & {} 0. \end{aligned}$$

Therefore, $F(t)\ge 0$ for $t\in [0,\infty )$. In particular, $F(0)\ge 0$, which implies

$$\begin{aligned} \int _{S^1}\log \frac{h_L}{h_K}\text{d}\bar{V}_K \ge \frac{1}{2} \log \frac{V(L)}{V(K)}. \end{aligned}$$

If the equality holds in (34), then $F(0)=0$, which implies $F(t)\equiv 0$ for $t\in [0,\infty )$. Therefore, $F'(t)\equiv 0$ for all $t\in [0,\infty )$. By Lemma 3.1, we have K and $L+tK$ are dilates, or K and $L+tK$ are parallelograms with parallel sides. So, K and L are dilates, or K and L are parallelograms with parallel sides. Conversely, if K and L are dilates, or K and L are parallelograms with parallel sides, the equality of (34) holds. $\square$

Remark 3.3

In Ref. [23], Xi and Leng proved that Theorems 1.5 and 1.6 are equivalent.

4 Proofs of Theorems 1.7 and 1.8

Suppose $K,L\in \mathcal {K}^n_o$. For $\phi \in \Phi$, the $\phi$-mixed volume $V_{\phi }(K,L)$ was defined in Ref. [17] by

$$\begin{aligned} V_{\phi }(K,L)=\int _{S^{n-1}}\phi \left(\frac{h_L}{h_K}\right )\text{d}V_K. \end{aligned}$$

(37)

The normalized $\phi$-mixed volume $\bar{V}_{\phi }(K,L)$ of $K,L\in \mathcal {K}^n_o$ was defined in Ref. [17] by

$$\begin{aligned} \bar{V}_{\phi }(K,L)=\phi ^{-1}\left(\int _{S^{n-1}}\phi \left(\frac{h_L}{h_K}\right )d\bar{V}_K\right ). \end{aligned}$$

(38)

In particular, if $\phi (t)=t^p$ with $p>0$, the normalized $\phi$-mixed volume $\bar{V}_{\phi }(K,L)$ reduces to the normalized $L_p$-mixed volume $\bar{V}_p(K,L)$.

We repeat the statements of Theorems 1.7 and 1.8.

Theorem 4.1

Suppose that $\phi \in \Phi$ with $\phi \ne \alpha \log (\alpha >0)$, and $K,L\in \mathcal K^2_o$ with $o\in K\cap L$. If K and L are at a dilation position, then

$$\begin{aligned} \int _{S^1}\phi \left(\frac{h_L}{h_K}\right )\text{d}\bar{V}_K\ge \phi \left( \frac{V(L)^{\frac{1}{2}}}{V(K)^{\frac{1}{2}}}\right) , \end{aligned}$$

(39)

with equality if and only if K and L are dilates.

Proof

From the log-concavity of $\phi ^{-1}$, we have

$$\begin{aligned} \int _{S^{n-1}}\log \frac{h_L}{h_K}\text{d}\bar{V}_K\le \log \circ \phi ^{-1}\left( \int _{S^{n-1}}\phi \left( \frac{h_L}{h_K}\right )\text{d}\bar{V}_K\right) , \end{aligned}$$

(40)

which is equivalent to

$$\begin{aligned} \exp \left( \int _{S^{n-1}}\log \frac{h_L}{h_K}\text{d}\bar{V}_K\right) \le \phi ^{-1}\left( \int _{S^{n-1}}\phi \left( \frac{h_L}{h_K}\right) \text{d}\bar{V}_K\right) . \end{aligned}$$

(41)

That is

$$\begin{aligned} \bar{V}_0(K,L)\le \bar{V}_{\phi }(K,L), \end{aligned}$$

(42)

with equality if and only if $h_L/h_K$ is constant on $\text {supp}S_K$. From (14), we have

$$\begin{aligned} \bar{V}_{\phi }(K,L)\ge \frac{V(L)^{\frac{1}{2}}}{V(K)^{\frac{1}{2}}}, \end{aligned}$$

(43)

which leads to (39). From the equality condition of (14) and (42), we have equality holds in (39) if and only if K and L are dilates. $\square$

Theorem 4.2

Suppose that $\phi \in \Phi$, $\phi \ne \alpha \log (\alpha >0)$ be concave on $(0,\infty )$, and $K,L\in \mathcal K^2_o$ with $o\in K\cap L$. If K and L are at a dilation position, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_{\phi }\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(44)

with equality for $\lambda \in (0,1)$ if and only if $K=L$.

Proof

Set $Q_{\lambda }=(1-\lambda )\cdot K+_{\phi } \lambda \cdot L$. From (16) and the concavity of $\phi$, we have

$$\begin{aligned} \phi (1)=(1-\lambda )\phi \left(\frac{h_K(u)}{h_{\lambda }(u)}\right)+\lambda \phi \left(\frac{h_L(u)}{h_{\lambda }(u)}\right)\le \phi \left( \frac{(1-\lambda ) h_K+\lambda h_L}{h_{\lambda }}\right) . \end{aligned}$$

(45)

By the monotone property of $\phi$, we have

$$\begin{aligned} h_{\lambda }\le (1-\lambda ) h_K+\lambda h_L. \end{aligned}$$

(46)

From (17), we have $h_{\lambda }=h_{Q_{\lambda }}$ with respect to the surface area measure $S_{Q_{\lambda }}$. Hence, we have

$$\begin{aligned} Q_{\lambda }\subset (1-\lambda )K+\lambda L. \end{aligned}$$

(47)

On the other hand, from (16), we have

$$\begin{aligned} 1=\phi ^{-1}\left( (1-\lambda )\phi \left(\frac{h_K(u)}{h_{\lambda }(u)}\right)+\lambda \phi \left(\frac{h_L(u)}{h_{\lambda }(u)}\right)\right) . \end{aligned}$$

(48)

From the log-concavity of $\phi$, we have

$$\begin{aligned} 0&= (\log \circ \phi ^{-1})\left((1-\lambda )\phi \left(\frac{h_K(u)}{h_{\lambda }(u)}\right)+\lambda \phi \left(\frac{h_L(u)}{h_{\lambda }(u)}\right)\right )\\&\ge (1-\lambda )\log \frac{h_K(u)}{h_{\lambda }(u)}+\lambda \log \frac{h_L(u)}{h_{\lambda }(u)}\\&= \log \frac{h_K^{1-\lambda }h_L^{\lambda }}{h_{\lambda }}, \end{aligned}$$

which implies $h_K^{1-\lambda }h_L^{\lambda }\le h_{\lambda }$. Hence,

$$\begin{aligned} (1-\lambda )\cdot K+_0\lambda \cdot L\subset Q_{\lambda }. \end{aligned}$$

(49)

From (13), we have

$$\begin{aligned} V(Q_{\lambda })\ge V((1-\lambda )\cdot K+_0\lambda \cdot L)\ge V(K)^{1-\lambda }V(L)^{\lambda }. \end{aligned}$$

(50)

If equality holds in (44), then $V((1-\lambda )\cdot K+_0\lambda \cdot L)= V(K)^{1-\lambda }V(L)^{\lambda }$. By the equality condition of (13), we have K and L are dilates. In addition, from $V(Q_{\lambda })= V((1-\lambda )\cdot K+_0\lambda \cdot L)$, we have $(1-\lambda )\cdot K+_0\lambda \cdot L= Q_{\lambda }$, which implies $K=L$. $\square$

We can get the $L_p$-Minkowski inequality and $L_p$-Brunn–Minkowski inequality for general planar convex bodies by setting $\phi (t)=t^p$ in Theorems 4.1 and 4.2.

Corollary 4.3

Suppose that $0<p<1$, and $K,L\in \mathcal K^2_o$ with $o\in K\cap L$. If K and L are at a dilation position, then

$$\begin{aligned} \left( \int _{S^1}\left( \frac{h_L}{h_K}\right) ^p \text{d}\bar{V}_K \right) ^{\frac{1}{p}}\ge \left( \frac{V(L)}{V(K)}\right) ^{\frac{1}{2}}, \end{aligned}$$

(51)

with equality if and only if K and L are dilates.

Corollary 4.4

Suppose that $0<p<1$, and $K,L\in \mathcal K^2_o$ with $o\in K\cap L$. If K and L are at a dilation position, then for all real $\lambda \in [0,1]$,

$$\begin{aligned} V\left((1-\lambda )\cdot K+_{\phi }\lambda \cdot L\right ) \ge V(K)^{(1-\lambda )} V(L)^{\lambda }, \end{aligned}$$

(52)

with equality for $\lambda \in (0,1)$ if and only if $K=L$.

5 $\phi$-Minkowski measure of asymmetry

In the well-known paper [6], abstracting from some extremal problems arising from geometry or other mathematical branches and from the previous work of many mathematicians, Grünbaum formulated a concept of measures of asymmetry (or symmetry) for convex bodies which, among other applications, can be used to describe how far a convex set is from a (centrally) symmetric one. Since then, the properties and applications of these known asymmetry measures are studied by many mathematicians (see [7,8,9,10,11, 22] and references therein).

In Ref. [7], Guo introduced a family of measures of (central) asymmetry, the so-called p-measures of asymmetry, which have the well-known Minkowski measure of asymmetry as a special case, and showed some similar properties of the p-measures to the Minkowski one. In Ref. [11], Jin, Leng and Guo extended the p-Minkowski measure of asymmetry to an Orlicz version. In addition, Jin et al. [11] showed that $p \;(1\le p\le \infty )$-Minkowski measures of asymmetry are closely related to $L_p$-mixed volumes. More precisely, we can define $p \;(1\le p\le \infty )$-Minkowski measures of asymmetry by $L_p$-mixed volumes. In Ref. [9], Jin introduced a measure of asymmetry $\text {as}_0(K)$ for planar convex bodies K in terms of the log-mixed volume, and extended the p-Minkowski measures of asymmetry to the case $0\le p\le \infty$.

For $K \in \mathcal {K}^n$, $x\in \text {int}(K)$ and $1 \le p\le \infty$, the p-Minkowski measure of asymmetry of K is defined by

$$\begin{aligned} \text {as}_p(C)=\inf _{x \in \text {int}(C)}\bar{V}_p(K_x, -K_x), \end{aligned}$$

(53)

where $K_x$ denotes $K+\{-x\}$. A point $x \in \text {int}(K)$ satisfying $\bar{V}_p(K_x, -K_x)=\text {as}_p(K)$ is called a p-critical point of K. The set of all p-critical points is denoted by $\mathcal {C}_p(K)$. The well-known Minkowski measure of asymmetry is the special case that $p=\infty$.

Theorem 5.1

([6, 7]) For $1 \le p \le \infty$, if $K \in \mathcal {K}^n$ then,

$$\begin{aligned} 1 \le \text {as}_p(K) \le n, \end{aligned}$$

(54)

equality holds on the left-hand side if and only if K is symmetric, and on the right-hand side if and only if K is a simplex.

For the p-critical set $\mathcal {C}_p(K)$, we have the following theorem.

Theorem 5.2

([6, 7]) For $1 \le p \le \infty$, and $K \in \mathcal {K}^n$, we have the following statements:

(1) if $p=1$, then $\mathcal {C}_1(K)=\text {int}(K)$;

(2) if $p=\infty$, then $\mathcal {C}_{\infty }(K)$ is a convex set with $\dim (\mathcal {C}_{\infty }(K))+\text {as}_{\infty }(K)\le n$;

(3) if $p\in (1,\infty )$, then $\mathcal {C}_{p}(K)$ is a singleton.

Note that if $K\in \mathcal {K}^2$, then $\mathcal {C}_{\infty }(K)$ is a singleton, i.e., each planar convex body has a unique critical $\infty$-critical point.

For fixed $K\in \mathcal {K}^n$, we denotes the unique p-critical point of K by $x_p$ for $p\in (1,\infty )$. It is easy to see that $x_p$ are coincide with the center of K if K is symmetric; if K is a simplex, then $x_p$ are coincide with the centroid of K. There are some other convex bodies that have this property that all $p(1<p<\infty )$-critical points coincide.

Example 5.3

(1) If $K:=a_1a_2a_3a_4$ with $a_1(-3,0),a_2(0,-3),a_3(4,0)$ and $a_4(0,3)$, then the quadrilateral K has centroid $c(\frac{1}{4},0)$ and $x_p(\frac{4}{15},0)$ for $p\in (1,\infty ]$;

(2) If $K:=a_1a_2a_3a_4$ with $a_1(-5,0),a_2(0,-5),a_3(12,0)$ and $a_4(0,5)$, then the quadrilateral K has centroid $c(\frac{7}{3},0)$ and $x_p(\frac{84}{41},0)$ for $p\in (1,\infty ]$.

Therefore, we state the following problem.

Problem 5.4

Suppose that $K \in \mathcal {K}^n$. Is it that $\dim (\text {conv}\{x_p: p\in (1,\infty )\})=0$?

The p-Minkowski measure of asymmetry for the case $p\in [0,1)$ is introduced in Ref. [9].

Given $K\in \mathcal {K}^2$, let $s \in \mathcal {C}_{\infty }(K)$ be the unique $\infty$-critical point of K. The log-Minkowski measure $\text {as}_0(K)$ of K is defined by

$$\begin{aligned} \text {as}_0(K)=\bar{V}_0(K_s, -K_s). \end{aligned}$$

(5.3)

Theorem 5.5

([9])

If $K\in \mathcal K^2$, then,

$$\begin{aligned} 1 \le \text {as}_0(K) \le 2. \end{aligned}$$

(56)

Equality holds on the left-hand side if and only if K is symmetric, and equality holds on the right-hand side if and only if K is a triangle.

If we define $\text {as}_0(K)=\inf _{x\in \text {int}(K)}\bar{V}_0(K_x, -K_x)$, then when K is a square, $\text {as}_0(C)<1$. This result shows that $\text {as}_0(K)$ is not a measure of asymmetry in the sense of Grünbaum [6].

In the following, we introduce a new measure of asymmetry in terms of the normalized $\phi$-mixed volume.

Definition 5.6

Suppose that $\phi \in \Phi$ be concave on $(0,\infty )$, $K\in \mathcal {K}^2$, and $s \in \mathcal {C}_{\infty }(K)$ be the unique $\infty$-critical point of K. The $\phi$-Minkowski measure $\text {as}_{\phi }(K)$ of K is defined by

$$\begin{aligned} \text {as}_{\phi }(K)=\bar{V}_{\phi }(K_s, -K_s). \end{aligned}$$

(57)

For the $\phi$-Minkowski measure, we have the following theorem.

Theorem 5.7

Suppose that $\phi \in \Phi$ be concave on $(0,\infty )$. If $K\in \mathcal K^2$, then,

$$\begin{aligned} 1 \le \text {as}_{\phi }(K) \le 2. \end{aligned}$$

(58)

Equality holds on the left-hand side if and only if K is symmetric, and equality holds on the right-hand side if and only if K is a triangle.

Proof

From (57), (42) and (56), we have

$$\begin{aligned} \text {as}_{\phi }(K)=\, & {} \bar{V}_{\phi }(K_s, -K_s)\\\ge \,& {} \bar{V}_0(K_s, -K_s)\\=\, & {} \text {as}_0(K)\\\ge\, & {} 1. \end{aligned}$$

On the other hand, from the concavity of $\phi$, we have

$$\begin{aligned} \int _{S^{n-1}}\phi \left( \frac{h_{-K_s}}{h_{K_s}}\right) \text{d}\bar{V}_{K_s}\le \phi \left( \int _{S^{n-1}}\frac{h_{-K_s}}{h_{K_s}}\text{d}\bar{V}_{K_s}\right) . \end{aligned}$$

(59)

From (27), (38), (53), (54) and (59), we have

$$\begin{aligned} \text {as}_{\phi }(K)=\, & {} \bar{V}_{\phi }(K_s, -K_s)\\=\, & {} \phi ^{-1}\left( \int _{S^{n-1}}\phi \left( \frac{h_{-K_s}}{h_{K_s}}\right) \text{d}\bar{V}_{K_s}\right) \\\le\, & {} \int _{S^{n-1}}\frac{h_{-K_s}}{h_{K_s}}\text{d}\bar{V}_{K_s}\\=\, & {} \bar{V}_1(K_s, -K_s)\\=\, & {} \text {as}_1(K)\\\le\, & {} 2. \end{aligned}$$

Hence,

$$\begin{aligned} 1\le \text {as}_0(K)\le \text {as}_{\phi }(K)\le \text {as}_1(K)\le 2. \end{aligned}$$

If K is symmetric, then we have $1= \text {as}_0(K)\le \text {as}_{\phi }(K)\le \text {as}_1(K)=1$, which implies $\text {as}_{\phi }(K)=1$; Conversely, if $\text {as}_{\phi }(K)=1$, then $1\le \text {as}_0(K)\le \text {as}_{\phi }(K)=1$, which implies $\text {as}_0(K)=1$, so K is symmetric.

If K is a triangle, then we have $2= \text {as}_0(K)\le \text {as}_{\phi }(K)\le \text {as}_1(K)=2$, which implies $\text {as}_{\phi }(K)=2$; Conversely, if $\text {as}_{\phi }(K)=2$, then $2= \text {as}_{\phi }(K)\le \text {as}_1(K)\le 2$, which implies $\text {as}_1(K)=2$, so K is a triangle. $\square$