Key words

1 Introduction

Besides the intrinsic interest in affine invariants originating in Felix Klein’s Erlangen Program, the extension to the Brunn-Minkowski-Firey theory [20, 21], and very recent connections between affine invariants and fields like stochastic geometry [3, 7] and information theory [17, 27, 30], led to an intense activity in this area of geometric analysis. The renewed interest in affine invariants has benefited also from a systematic approach classifying them, as for example in [8, 15, 16, 18], and from their use in affine and affine Sobolev inequalities [10, 11, 19, 23–26, 28, 39, 40] and problems arising in differential geometry [4–6, 9, 22, 35–38] which rely on isoperimetric-type functional inequalities. The study of such inequalities constitutes one of our primary goals of an on-going project.

The present paper spun as a follow-up of [34] in which we introduced new SL(n)-invariants for smooth convex bodies. We started by searching for sharp affine inequalities satisfied by one such invariant derived, in a certain sense, from the centro-affine surface area. The resulting inequalities are the subject of the next section. In the process, we encountered a connection to another SL(n) invariant of convex bodies defined by Paouris and Werner who also related it to information theory [30]. In Sect. 3, we present two alternative definitions of this invariant. We noted that an additional SL(n) invariant of convex bodies of class C  +  2 is defined with analogous techniques. This prompted us to conjecture that SL(n) invariants for convex bodies with continuous and positive centro-affine curvature function can be obtained as limits of normalized p-affine surface areas of the convex body.

The setting for this paper is the Euclidean space ℝn,  n ≥ 2, in which we consider convex bodies containing the origin in their interior. Most of the time, we will also require that the convex bodies have smooth boundary, i.e. C ∞, with positive Gauss curvature. We will denote the set of such convex bodies by \(\mathcal{K}_{\mbox{ reg}}\). However, on several occasions, we will relax the regularity of the boundary to class C 2 with positive Gauss curvature and we will use the notation C  +  2 to indicate this latter class of convex bodies. The preferred parametrization of a convex body K will be with respect to the unit normal vector, u↦X K (u), making many functions on the boundary ∂K to be considered as functions on the unit sphere \({\mathbb{S}}^{n-1}\).

We will denote the Gauss curvature of a convex body by \(\mathcal{K}\) and its centro-affine curvature by \(\mathcal{K}_{0}\). Geometrically, \(\mathcal{K}_{0}^{-1/2}\) at a given point of ∂K is, up to a dimension dependent constant, the volume of the centered osculating ellipsoid at that point. Note that the centro-affine curvature is constant if and only if K is a centered ellipsoid. This can also be seen from a lemma due to Petty [31] since, analytically, as a function on the unit sphere, the centro-affine curvature is the ratio \(\mathcal{K}_{0}(u) = \frac{\mathcal{K}(u)} {{h}^{n+1}(u)},\ u \in {\mathbb{S}}^{n-1}\), where h is the support function of K: h(u) = max{x ⋅u∣x ∈ K} with x ⋅u denoting the usual inner product in ℝn. Two additional notations are deemed necessary. First, \(\mathcal{N}_{0}(u) :={ \mathcal{K}_{0}}^{- \frac{1} {n+1} }(u)\:\mathcal{N}(u)\) is the centro-affine normal which is, pointwise, proportional to the (classical) affine normal \(\mathcal{N}(u)\), [14]. Finally, we will use dμ K to denote the cone measure of ∂K which, given that the Gauss curvature of K is positive, can be expressed by \(d\mu _{K}(x) = h(\nu (x))\, \frac{1} {\mathcal{K}}(\nu (x))\,d\mu _{{\mathbb{S}}^{n-1}}(\nu (x))\), where \(\nu : \partial K \rightarrow {\mathbb{S}}^{n-1}\) is the Gauss map of the boundary of K, hence the inverse of the parametrization X.

2 Inequalities for a Second Order Centro-Affine Invariant

In [34], we introduced a class of SL(n) invariants for smooth convex bodies in ℝn. For a fixed convex body K, these invariants were the first, second, and, for an arbitrary integer k, the k-th variation of the volume of K while the boundary of the body was subject to a pointwise deformation in the direction of the centro-affine normal by a speed equal to a power of the centro-affine curvature at each specific point. The p-affine surface areas introduced by Lutwak [21] for p greater than one (later extended to the range 0 ≤ p < 1 by Hug [13], to − n ≤ p < 0 by Meyer-Werner [29], and to − ∞ ≤ p < n by Schütt-Werner [33]) are, via this method, part of this class of invariants. To exemplify, and also bring the reader’s attention to a particular such invariant which is one of the main objects of this paper, let us consider the following deformation of a convex body K with smooth boundary:

$$\left \{\begin{array}{rlrl}2\frac{\partial X(u,t)} {\partial t} & =& {\mathcal{K}_{0}}^{\frac{1} {2} }(u,t)\:\mathcal{N}_{0}(u,t)& \cr X(u,0)& =& \ X_{K}(u).& \cr \end{array}\right.$$
(1)

Then, the first variation of Vol(K) is the centro-affine surface area of K:

$$\frac{d} {dt}\left (V ol(K)\right )_{t=0} = -\displaystyle\int _{\partial K}\mathcal{K}_{0}^{\frac{1} {2} }(\nu (x))\,d\mu _{K}(x) = -\Omega _{n}(K) =: \Omega _{1,n}(K),$$
(2)

see [34]. Recall that the centro-affine surface area of a convex body is the only one among the p-affine surface areas, \(\Omega _{p}(K) =\int _{\partial K}\mathcal{K}_{0}^{ \frac{p} {n+p} }\,d\mu _{K},\) invariant under GL(n) transformations of the Euclidean space. Moreover, pursuing an additional variation, we obtain:

$$\displaystyle\begin{array}{lll} \Omega _{2,n}(K)&& \\ & & \,\,\,:= -\left (\frac{{d}^{2}\,V ol(K(t)} {d{t}^{2}} \right )_{\mid _{t=0}}\end{array}$$
(3)
$$\displaystyle\begin{array}{lll} & & \,\,\,= \frac{n(n - 1)} {2} \,V ol({K}^{\circ }) -\frac{n - 1} {2} \,\displaystyle\int _{{\mathbb{S}}^{n-1}}h\sqrt{\mathcal{K}_{0}}\,s(h\sqrt{\mathcal{K}_{0}},h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}}, \\ \end{array}$$

where s(f 1, f 2, ⋯ , f n − 1) is an extension of the mixed curvature function usually defined on C 2, here smooth, support functions to arbitrary smooth functions on the unit sphere \({\mathbb{S}}^{n-1}\), see [32] page 115 and also [34]. For the reader familiar with mixed determinants, the following can be taken as definition for the function \(s(f_{1},f_{2},\cdots \,,f_{n-1})(u) := D(((f_{1})_{ij}+\delta _{ij}f_{1})(u),((f_{2})_{ij}+\delta _{ij}f_{2})(u),\ldots, ((f_{n-1})_{ij}+\delta _{ij}f_{n-1})(u)),u \in {\mathbb{S}}^{n-1},\)where D is a mixed determinant and ( . ) i represents the covariant differentiation with respect to the i-th vector of a positively oriented orthonormal frame on the unit sphere \({\mathbb{S}}^{n-1}\).

We will show in Proposition 1 that, in a certain sense, Ω 2, n (K) measures how far K is from being a centered ellipsoid. In preparation, we call the Aleksandrov body, A f , associated with a continuous positive function f on the unit sphere the convex body whose support function h f is the maximal element of

$$\{h \leq f\mid h : {\mathbb{S}}^{n-1} \rightarrow \mathbb{R}\ \mbox{ support function of a convex body}\}.$$

If f is itself a support function of a convex body L, then A f is precisely the body L. Moreover, in general, f = h f almost everywhere with respect to the surface area measure of A f . We could not find where this notion first surfaced in the literature, yet the work [9] gives an excellent background on this notion. We are now ready to state the following comparison result which we will use in analyzing Ω 2, n :

Lemma 1 (Monotonicity Lemma). 

Suppose that f is a strictly positive smooth function on the unit sphere \({\mathbb{S}}^{n-1}\) and that h is the support function of a convex body K ⊂ ℝ n which belongs to \(\mathcal{K}_{\mbox{ reg}}\) . Then, denoting by \(m :=\min _{{\mathbb{S}}^{n-1}} \frac{f} {h}\) , respectively, \(M :=\max _{{\mathbb{S}}^{n-1}} \frac{f} {h}\) , we have

$$m \cdot n\,V ol(K) \leq \displaystyle\int _{{\mathbb{S}}^{n-1}}fs(h,h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}} \leq M \cdot n\,V ol(K)$$
(4)

and, if the Aleksandrov body associated with f has continuous positive curvature function, then

$${m}^{2} \cdot n\,V ol(K) \leq \displaystyle\int _{{ \mathbb{S}}^{n-1}}fs(f,h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}} \leq {M}^{2} \cdot n\,V ol(K).$$
(5)

Proof.

Since K belongs to \(\mathcal{K}_{\mbox{ reg}}\), s(h, h, …, h) > 0 on \({\mathbb{S}}^{n-1}\), thus mh ≤ f ≤ Mh implies directly (4). In fact, we will show that we also have

$$m \cdot V (h,g,h,\ldots, h) \leq V (f,g,h,\ldots, h) \leq M \cdot V (h,g,h,\ldots, h),$$
(6)

for any g support function of a convex body, denoted for later use by K 2. Indeed, if f itself would be a support function of a convex body, this claim is simply due to the monotonicity of mixed volumes. If f is not a support function, then there exists a large enough constant c so that f + ch is a support function of a convex body, say L, with the Gauss parametrization. Moreover, L ⊆ K 1, where the latter is the dilation of K by the factor M + c. Then, from the monotonicity of mixed volumes, we have that V (L, K 2, K, …, K) ≤ V (K 1, K 2, K, …, K). Choosing to represent these mixed volumes through the notation emphasizing the support functions of the two convex bodies, we have V (f + ch, g, h, …, h) ≤ V ((M + c)h, g, h, …, h). Finally, using the linearity of mixed volumes, we obtain V (f, g, h, …, h) + cV (h, g, h, …. , h) ≤ (M + c)V (h, g, h, …, h) which is, after a trivial simplification, the right-hand side inequality of Eq. (6). Similarly, by considering the dilation K of factor (m + c), we obtain a convex body K 3 such that K 3 ⊆ L and an argument analogous with the one above will imply mV (h, g, h, …, h) ≤ V (f, g, h, …, h).

We will now proceed to prove Eq. (5). Note again that if f would be a support function of a convex body, the claim follows from the monotonicity of mixed volumes. If f is not a support function, consider the Aleksandrov body associated to f, A f , whose support function we denote by h f . Thus Mh ≥ f ≥ h f  ≥ mh and, \(S_{A_{f}}\)-a.e., \(f \circ \nu _{A_{f}}(x) = h_{f}(x)\), where \(\nu _{A_{f}}\) is the Gauss map of ∂A f . As, by hypothesis, A f has a continuous positive curvature function, and by using Eq. (6), we have

$$\displaystyle\begin{array}{rcl} \displaystyle\int _{{\mathbb{S}}^{n-1}}fs(f,h,...,h)\,d\mu _{{\mathbb{S}}^{n-1}}& =& \displaystyle\int _{\partial A_{f}}f(\nu _{A_{f}}^{-1}(x))s(f,h,\ldots, h)(\nu _{ A_{f}}^{-1}(x))\,dS_{ A_{f}}(x) \\ & =& \displaystyle\int _{\partial A_{f}}h_{f}(\nu _{A_{f}}^{-1}(x))s(f,h,\ldots, h)(\nu _{ A_{f}}^{-1}(x))\,dS_{ A_{f}}(x) \\ & =& \displaystyle\int _{{\mathbb{S}}^{n-1}}h_{f}s(f,h,...,h)\,d\mu _{{\mathbb{S}}^{n-1}} \\ & =& nV (f,h_{f},h,...,h) \\ & \geq & m \cdot n\,V (h,h_{f},h,\ldots, h) \\ & =& m\displaystyle\int _{{\mathbb{S}}^{n-1}}h_{f}s(h,h,...,h)\,d\mu _{{\mathbb{S}}^{n-1}} \\ & \geq & m\displaystyle\int _{{\mathbb{S}}^{n-1}}m\,hs(h,h,...,h)\,d\mu _{{\mathbb{S}}^{n-1}} \\ & =& {m}^{2} \cdot n\,V ol(K). \end{array}$$
(7)

The second inequality can be proved similarly.

Consequently, we obtain the following inequalities for Ω 2, n (K).

Proposition 1.

Let \(K \in \mathcal{K}_{\mbox{ reg}}\) with the usual notations of h and \(\mathcal{K}_{0}\) for the support function, respectively, the centro-affine curvature of K as functions on the sphere \({\mathbb{S}}^{n-1}\) . Then

  1. 1.

    Ω n,2 (K) ≥ 0 with equality if and only if K is a centered ellipsoid.

  2. 2.

    If, in addition, the Aleksandrov body associated with \(f := h\sqrt{\mathcal{K}_{0}}\) has continuous positive curvature function, then \(\Omega _{n,2}(K) \leq \frac{(n - 1)n} {2} (M - m)V ol(K)\) , where M,m are the maximum and minimum of the centro-affine curvature of K. The equality occurs if and only if K is a centered ellipsoid.

Proof.

  1. 1.

    The first claim follows immediately from the Minkowski-type inequality we detailed in Lemma 4.3 of [34]

    $$\displaystyle\begin{array}{rcl} & & \left (\displaystyle\int _{{\mathbb{S}}^{n-1}}fs(f,h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}}\right )\left (\displaystyle\int _{{\mathbb{S}}^{n-1}}hs(h,h,\ldots, h)d\mu _{{\mathbb{S}}^{n-1}}\right ) \\ & & \qquad \leq {\left (\displaystyle\int _{{\mathbb{S}}^{n-1}}fs(h,h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}}\right )}^{2}, \\ \end{array}$$

    where f is an arbitrary smooth function on the sphere, while h is a smooth support function of a convex body. It suffices to apply this inequality to the second term of Ω n, 2(K) with \(f := h\sqrt{\mathcal{K}_{0}}\) to obtain

    $$\Omega _{2,n}(K) \geq \frac{n(n - 1)} {2} \,V ol({K}^{\circ }) -\frac{n - 1} {2n} \,\frac{\Omega _{n}^{2}(K)} {V ol(K)}$$

    from which the result follows by Hölder’s inequality

    $$\displaystyle\begin{array}{rcl} V ol({K}^{\circ }) \cdot V ol(K)& & \\ & & \,\,\,= \frac{1} {{n}^{2}}\left (\displaystyle\int _{\partial K}\mathcal{K}_{0}\,d\mu _{K}\right ) \cdot \left (\displaystyle\int _{\partial K}d\mu _{K}\right ) \geq \frac{1} {{n}^{2}}\,{\left (\displaystyle\int _{\partial K}\sqrt{\mathcal{K}_{0}}\,d\mu _{K}\right )}^{2}. \end{array}$$
    (8)

    Note that the equality is attained if and only if \(\mathcal{K}_{0}\) is constant on \({\mathbb{S}}^{n-1}\), hence if and only if K is a centered ellipsoid.

  2. 2.

    By taking \(f = h\sqrt{\mathcal{K}_{0}}\) with \(m \leq \mathcal{K}_{0} \leq M\), we can apply Eq. (5),

    $$\displaystyle\begin{array}{rcl} \Omega _{2,n}(K)& =& \frac{n(n - 1)} {2} \,V ol({K}^{\circ }) \\ & & -\frac{n - 1} {2} \,\displaystyle\int _{{\mathbb{S}}^{n-1}}h\sqrt{\mathcal{K}_{0}}\,s(h\sqrt{\mathcal{K}_{0}},h,\ldots, h)\,d\mu _{{\mathbb{S}}^{n-1}}, \\ & \leq & \frac{n(n - 1)} {2} \, \frac{1} {n}\displaystyle\int _{\partial K}\mathcal{K}_{0}\,d\mu _{K} -\frac{n(n - 1)} {2} \,m\,V ol(K) \\ & \leq & \frac{n(n - 1)} {2} \,\left (M - m\right )\,V ol(K). \end{array}$$
    (9)

    Equality is attained if and only if M = m which implies, as before, that K is a centered ellipsoid. Note that we have only used the left-hand side inequality of Eq. (5). It so happens that the right-hand side inequality of Eq. (5) follows for this choice of function f from the positivity of Ω 2, n (K) for any \(K \in \mathcal{K}_{\mbox{ reg}}\).

Further, the previous result implies additional isoperimetric-type inequalities.

Theorem 1.

If \(K \in \mathcal{K}_{\mbox{ reg}}\) , the following Gl(n)-invariant inequality holds

$$\displaystyle\begin{array}{rcl} \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K)& \leq & V ol(K) \cdot V ol({K}^{\circ }) \\ & \leq & \frac{2} {n(n - 1)}\min \{V ol(K) \cdot \Omega _{n,2}(K),V ol({K}^{\circ }) \cdot \Omega _{ n,2}({K}^{\circ })\} + \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K),\\ \end{array}$$

and equality occurs if and only if K is a centered ellipsoid.

If, in addition, K is such that the Aleksandrov body associated with \(f := h\sqrt{\mathcal{K}_{0}}\) has continuous positive curvature function and \(\frac{M} {m} \leq \frac{1 + \sqrt{5}} {2}\) , the golden ratio, then the following Gl(n)-invariant inequality holds:

$$\frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K) \leq V ol(K) \cdot V ol({K}^{\circ }) \leq \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K)\,{\left [1 -\frac{M - m} {\sqrt{Mm}} \right ]}^{-1}$$

with equality if and only if K is a centered ellipsoid.

Proof.

The left-hand inequality follows immediately from Hölder’s inequality. In fact, this easy remark motivated a search for an upper bound of the volume product Vol(K) ⋅Vol(K  ∘ ) in terms of the centro-affine surface area or, in other words, a reverse isoperimetric-type inequality.

Toward this goal, note that the sign of Ω 2, n (K) translates into the following Gl(n)-invariant inequality:

$$\frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K) \leq V ol(K) \cdot V ol({K}^{\circ }) \leq \frac{2} {n(n - 1)}V ol(K) \cdot \Omega _{n,2}(K) + \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K),$$

with equality if and only if K is a centered ellipsoid. Apply the same inequality with the roles of K and K  ∘  reversed and use the fact that Ω n (K) = Ω n (K  ∘ ), [12, 18, 39]. Therefore,

$$\displaystyle\begin{array}{rcl} \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K)& \leq & V ol(K) \cdot V ol({K}^{\circ }) \\ & \leq & \frac{2} {n(n - 1)}\min \{V ol(K) \cdot \Omega _{n,2}(K),V ol({K}^{\circ }) \cdot \Omega _{ n,2}({K}^{\circ })\} + \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K),\\ \end{array}$$

with equality if and only if K is a centered ellipsoid.

From Proposition 1,

$$\frac{2} {n(n - 1)}\,V ol(K) \cdot \Omega _{n,2}(K) \leq (M - m)\,V o{l}^{2}(K)$$

and

$$\frac{2} {n(n - 1)}\,V ol({K}^{\circ }) \cdot \Omega _{ n,2}({K}^{\circ }) \leq ({M}^{\circ }- {m}^{\circ })\,V o{l}^{2}({K}^{\circ }),$$

thus

$$\displaystyle\begin{array}{rcl} & & \frac{2} {n(n - 1)}\min \{V ol(K) \cdot \Omega _{n,2}(K),V ol({K}^{\circ }) \cdot \Omega _{ n,2}({K}^{\circ })\} \\ & & \qquad \leq \sqrt{(M - m)({M}^{\circ } - {m}^{\circ } )}\,V ol(K) \cdot V ol({K}^{\circ }).\end{array}$$

Here m  ∘  and M  ∘  are the minimum, respectively, the maximum of the centro-affine curvature of ∂K  ∘ .

For any point of ∂K, x, there exists a point y on ∂K  ∘  such that \(\mathcal{K}_{0}(x) \cdot \mathcal{K}_{0}^{\circ }(y) = 1\), see [12], thus M ⋅m  ∘  = 1 and m ⋅M  ∘  = 1 otherwise a contradiction with one of the definitions of m  ∘ ,  M  ∘  occurs. Hence

$$\sqrt{(M - m)({M}^{\circ } - {m}^{\circ } )} = \sqrt{(M - m)\left ( \frac{1} {m} - \frac{1} {M}\right )} = \frac{M - m} {\sqrt{Mm}}, $$

which is less or equal to 1 if and only if M ∕ m is less or equal to the golden ratio above.

Thus

$$V ol(K) \cdot V ol({K}^{\circ }) \leq \frac{M - m} {\sqrt{Mm}} \cdot V ol(K) \cdot V ol({K}^{\circ }) + \frac{1} {{n}^{2}}\,\Omega _{n}^{2}(K)$$

which implies the right-hand side inequality. The equalities follow as before from M = m equivalent to constant centro-affine curvature along the boundary ∂K.

Note that in the next proposition we drop the smoothness assumption on the boundary of K to class C 2.

Proposition 2.

For any p > 1, and any K ∈ C + 2 with the origin in its interior, we have

$$\frac{\Omega _{p}^{n+p}(K)} {V o{l}^{n-p}(K)} \leq {n}^{p-1}\,{\left (V ol(K) \cdot V ol({K}^{\circ })\right )}^{p-1} \cdot \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)}.$$
(10)

The equality holds if and only if p = 1 or K is a centered ellipsoid.

The opposite inequality holds for p < 1, p≠ − n.

Proof.

Note that, for any p≠ − n,

$$\Omega _{p}(K) =\displaystyle\int _{\partial K}\mathcal{K}_{0}^{ \frac{p} {n+p} }\,d\mu _{K} =\displaystyle\int _{\partial K}{\left (\mathcal{K}_{0}^{ \frac{n} {n+1} }\right )}^{\frac{p-1} {n+p} }\,d\sigma _{K},$$
(11)

where dσ K is the affine surface area measure, in other words the Blaschke metric, of K. As the function \(x\mapsto {x}^{\frac{p-1} {n+p} },\ x > 0\), is concave for p ≥ 1 and convex for p ≤ 1, we apply the appropriate Jensen’s inequality for each range and the normalized measure \(\frac{1} {\Omega (K)}\,d\sigma _{K}\). If p ≥ 1, we obtain

$${ \left (\frac{nV ol({K}^{\circ })} {\Omega (K)} \right )}^{\frac{p-1} {n+p} } \geq \frac{\Omega _{p}(K)} {\Omega (K)} \ \ \ \Leftrightarrow \ \ \ \Omega _{p}(K) \leq {\left (n\,V ol({K}^{\circ })\right )}^{\frac{p-1} {n+p} } \cdot {\Omega }^{\frac{n+1} {n+p} }(K)\,\,\,$$
(12)

with equality if and only if p = 1 or K is a centered ellipsoid. A re-arrangement of terms, gives Eq. (11). The proof of the reverse inequality in the case p ≤ 1 is perfectly similar.

Corollary 1.

For any convex body \(K \in \mathcal{K}_{\mbox{ reg}}\),

$$\displaystyle\begin{array}{lll}{ n}^{n}\left [ \frac{2} {n - 1}V ol(K) \cdot \Omega _{n,2}(K) + \frac{1} {n}\,\Omega _{n}^{2}(K)\right ]&& \\ \qquad\geq \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)} \geq \frac{\Omega _{n}^{2n}(K)} {{[(2/(n - 1))\,\Omega _{n,2}(K)V ol(K) + \Omega _{n}^{2}(K)/n]}^{n-1}}, \\ \end{array}$$

with equality iff K is a centered ellipsoid.

Proof.

Apply the previous result for p = 0 and, respectively, p = n, and use the bounds on Vol(K) ⋅Vol(K  ∘ ) from Theorem 1.

Corollary 2 (Isoperimetric-like Inequality). 

For any K ∈ C + 2 with the centroid at the origin, and any T ∈ Sl(n),

$$\frac{{S}^{n}(TK)} {V o{l}^{n-1}(K)} \geq \frac{n} {\omega _{n}^{2n-3}}\,\max \left \{ \frac{\Omega _{n}^{2n}(K)} {{\Omega }^{n+1}(K)/V o{l}^{n-1}(K)},{\left ( \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)}\right )}^{n-1}\right \},$$
(13)

where S(TK) stands for the surface area of TK and ω n is the volume of the unit ball x 1 2 + … + x n 2 = 1 in ℝ n . Equality occurs if and only if K is a centered ellipsoid and T is the linear transformation of determinant one such that TK is a ball.

Hence

Proof.

Consider p = n in the inequality of Proposition 2 to obtain

$$\Omega _{n}^{2n}(K) \leq {n}^{n-1}{[V ol(K) \cdot V ol({K}^{\circ })]}^{n-1} \cdot \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)}.$$
(14)

From the classical isoperimetric inequality,

$$V o{l}^{n-1}(K) \leq \left (V o{l}^{n-1}(B)/{S}^{n}(B)\right ){S}^{n}(K),$$

where B is the unit ball as above. On the other hand, by Blaschke-Santaló inequality, Vol(K) ⋅Vol(K  ∘ ) ≤ (Vol(B))2.

Therefore

$$\Omega _{n}^{2n}(K) \leq {n}^{n-1}\frac{V o{l}^{3(n-1)}(B)} {{S}^{n}(B)} \, \frac{{S}^{n}(K)} {V o{l}^{n-1}(K)} \cdot \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)},$$
(15)

where all quantities, except S(K), are invariant under linear transformations of determinant one. Hence, the conclusion follows as \({n}^{n-1}\,\frac{V o{l}^{3(n-1)}(B)} {{S}^{n}(B)} = \frac{\omega _{n}^{2n-3}} {n}.\) To analyze the equality case one needs to take T to be the linear transformation of determinant one minimizing the surface area of K and note that all other equalities hold if and only if K is a centered ellipsoid.

We will now use p = 0 in Proposition 2, to obtain

$$\displaystyle\begin{array}{rcl} \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)}& \leq & n\,V ol(K) \cdot V ol({K}^{\circ }) \leq n\, \frac{V (B)} {S{(B)}^{n/(n-1)}} \cdot S{(K)}^{n/(n-1)} \cdot V ol({K}^{\circ }) \\ & \leq & n\, \frac{V {(B)}^{3}} {S{(B)}^{n/(n-1)}} \cdot \frac{S{(K)}^{n/(n-1)}} {V ol(K)} = {n}^{1- \frac{n} {n-1} }\,\omega _{n}^{3- \frac{n} {n-1} }\, \frac{S{(K)}^{n/(n-1)}} {V ol(K)}, \\ & & \end{array}$$
(16)

relying again on Blaschke-Santaló inequality.

From here,

$$\frac{{S}^{n}(TK)} {V o{l}^{n-1}(K)} \geq \frac{n} {\omega _{n}^{2n-3}}\,{\left ( \frac{{\Omega }^{n+1}(K)} {V o{l}^{n-1}(K)}\right )}^{n-1},$$
(17)

with the same condition for the equality case as above.

One can use K. Ball’s reverse isoperimetric ratio which gives an upper bound on \(\frac{{S}^{n}(TK)} {V o{l}^{n-1}(K)}\) by the corresponding ratio for the regular solid simplex in ℝn (or the solid cube in the centrally-symmetric case), [1, 2], in the above corollary to get lower bounds on the affine isoperimetric ratio of bodies in C  +  2. However, these bounds will not be sharp.

As in Corollary 1, one can drop the requirement that the centroid of K is at the origin, consider \(K \in \mathcal{K}_{\mbox{ reg}}\), and use the upper bound on the volume product from Theorem 1 instead of Blaschke-Santaló inequality, to obtain SL(n) invariant lower bounds on the isoperimetric ratio S(TK)n ∕ Vol(K)n − 1.

Finally, we include the next corollary, due to [30], which follows immediately from Proposition 2.

Corollary 3.

For any convex body K of class C + 2 containing the origin in its interior,

$$\Omega _{K} \leq \frac{{\Omega }^{n+1}(K)} {{(nV ol({K}^{\circ }))}^{n+1},}$$
(18)

where \(\Omega _{K} :=\lim \limits_{p\rightarrow \infty }{\left ( \frac{\Omega _{p}(K)} {nV ol({K}^{\circ })}\right )}^{n+p}\) is the affine invariant introduced by Paouris and Werner in [30]. The equality occurs if and only if K is a centered ellipsoid.

Note that in [30], for certain considerations, the invariant Ω K has been defined for convex bodies whose centroid is at the origin, yet the above definition makes sense for any convex body K of class C  +  2 containing the origin in its interior for which one can show as in [30] that the limit exists.

3 More on the Paouris-Werner Invariant

Motivated by the earlier occurrence of Ω K , we would like to give here a couple of other definitions of this invariant when K belongs to C  +  2. To do so, let us also recall that Paouris and Werner showed in [30] that Ω K is related to the Kullback-Leibler divergence D KL of two specific probability measures P, Q on ∂K via the relation \(D_{KL}(P\|Q) =\ln \left ( \frac{V ol(K)} {V ol({K}^{\circ })}\Omega _{K}^{-1/n}\right ),\) where, in slightly different terms than in [30],

$$D_{KL}(P\|Q) := \frac{1} {nV ol({K}^{\circ })}\displaystyle\int _{\partial K}\mathcal{K}_{0}\ln \left (\mathcal{K}_{0}\, \frac{V ol(K)} {V ol({K}^{\circ })}\right )\,d\mu _{K}.$$

Hence, it is useful to note the identity

$$\ln (\Omega _{K}) = - \frac{1} {V ol({K}^{\circ })}\displaystyle\int _{\partial K}\mathcal{K}_{0}\ln \mathcal{K}_{0}\,d\mu _{K},$$
(19)

and note that, in this paper, we assume only that the origin is contained in the interior of the convex body K.

Proposition 3.

For any K of class C + 2 containing the origin in its interior, and any integer p > 1, the following Gl(n)-invariant inequalities hold

$$\Omega _{n}^{2}(K) \geq \frac{{\left (\Omega _{n/3}(K)\right )}^{4}} {{(nV ol(K))}^{2}} \geq \frac{{\left (\Omega _{n/7}(K)\right )}^{8}} {{(nV ol(K))}^{6}} \geq \ldots \geq \frac{{\left (\Omega _{n/({2}^{p}-1)}(K)\right )}^{{2}^{p} }} {{(nV ol(K))}^{{2}^{p}-2}} \geq \ldots, $$
(20)

or, alternately,

$$\Omega _{n}^{2}(K) \geq \frac{{\left (\Omega _{3n}({K}^{\circ })\right )}^{4}} {{(nV ol(K))}^{2}} \geq \frac{{\left (\Omega _{7n}({K}^{\circ })\right )}^{8}} {{(nV ol(K))}^{6}} \geq \ldots \geq \frac{{\left (\Omega _{n({2}^{p}-1)}({K}^{\circ })\right )}^{{2}^{p} }} {{(nV ol(K))}^{{2}^{p}-2}} \geq \ldots, $$
(21)
$$\Omega _{n}^{2}(K) \geq \frac{{\left (\Omega _{3n}(K)\right )}^{4}} {{(nV ol({K}^{\circ }))}^{2}} \geq \frac{{\left (\Omega _{7n}(K)\right )}^{8}} {{(nV ol({K}^{\circ }))}^{6}} \geq \ldots \geq \frac{{\left (\Omega _{n({2}^{p}-1)}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p}-2}} \geq \ldots.$$
(22)

In all sequences, all equalities hold if and only if K is a centered ellipsoid (which is the only reason why we did not include p = 1 in the statement).

Proof.

Note that Eqs. (21) and (22) are equivalent through the equality \(\Omega _{q}(K) = \Omega _{{n}^{2}/q}({K}^{\circ })\), [12, 18, 39]. The same goes for Eq. (23) due to Ω n (K) = Ω n (K  ∘ ) and interchanging the roles of K and K  ∘  in the previous sequence of inequalities. Thus, it suffices to prove (21).

We will use the concavity of the function \(x\mapsto \sqrt{x}\) on (0, ∞) and Jensen’s inequality as follows:

$${ \left ( \frac{\Omega _{n}(K)} {nV ol(K)}\right )}^{1/2} ={ \left (\displaystyle\int _{ \partial K}\sqrt{\mathcal{K}_{0}}\, \frac{d\mu _{K}} {n\,V ol(K)}\right )}^{1/2} \geq \displaystyle\int _{ \partial K}\root{4}\of{\mathcal{K}_{0}}\, \frac{1} {n\,V ol(K)}\,d\mu _{K},$$
(23)

thus

$${\left ( \frac{\Omega _{n}(K)} {nV ol(K)}\right )}^{1/2} \geq \frac{\Omega _{n/3}(K)} {nV ol(K)},$$

which is, after raising both sides to power four, the first inequality of Eq. (21). Re-iterate now the same argument for Ω n ∕ 3(K):

$${ \left (\frac{\Omega _{n/3}(K)} {nV ol(K)}\right )}^{1/2} ={ \left (\displaystyle\int _{ \partial K}\root{4}\of{\mathcal{K}_{0}}\, \frac{d\mu _{K}} {n\,V ol(K)}\right )}^{1/2} \geq \displaystyle\int _{ \partial K}\root{8}\of{\mathcal{K}_{0}}\, \frac{1} {n\,V ol(K)}\,d\mu _{K},$$
(24)

which translates into

$${\left (\frac{\Omega _{n/3}(K)} {nV ol(K)}\right )}^{1/2} \geq \frac{\Omega _{n/7}(K)} {nV ol(K)}.$$

Hence

$$\Omega _{n}(K) \geq \frac{\Omega _{n/3}^{2}(K)} {nV ol(K)} \geq \frac{\Omega _{n/7}^{4}(K)} {{(nV ol(K))}^{3}}$$

and so on, the sequence is obtained by iterating the argument.

Theorem 2 (Alternative Definition of Ω K ). 

For any K of class C + 2 containing the origin in its interior, the scaling invariant sequence

$$\left \{\frac{{\left (\Omega _{n({2}^{p}-1)}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p} }} \right \}_{p\in \mathbb{N},\ p\geq 1}$$

converges and

$$\lim _{p\rightarrow \infty }{\left (\frac{\Omega _{n({2}^{p}-1)}(K)} {nV ol({K}^{\circ })} \right )}^{{2}^{p} } = \Omega _{K}.$$
(25)

Proof.

By Eq. (23), the positive sequence \(\frac{{\left (\Omega _{n({2}^{p}-1)}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p}-2}}\) is decreasing, thus converges. Therefore, so does the sequence above whose general term differs from general term of the former sequence by a factor of (nVol(K  ∘ )) − 2.

Let q : = n(2p − 1), and, similarly with Proposition 3.6 in [30], consider

$$\displaystyle\begin{array}{lll} {}& {} &\ln \left [\lim _{p\rightarrow \infty }\frac{{\left (\Omega _{n({2}^{p}-1)}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p} }} \right ] \\ & & =\lim _{p\rightarrow \infty }{2}^{p}\ln \left (\frac{\Omega _{n({2}^{p}-1)}(K)} {nV ol({K}^{\circ })} \right ) \\ & & = -\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dp}\left (\Omega _{n({2}^{p}-1)}(K)\right )} {\Omega _{n({2}^{p}-1)}(K)} \\ & & = -\lim _{p\rightarrow \infty }\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dq}\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{ \frac{q} {n+q} }\right )\,d\mu _{K}\right )\frac{dq} {dp}} {\Omega _{n({2}^{p}-1)}(K)} \\ & & = -\lim _{p\rightarrow \infty }{2}^{2p}\,\frac{ \frac{d} {dq}\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{ \frac{q} {n+q} }\right )\,d\mu _{K}\right )} {\Omega _{n({2}^{p}-1)}(K)} \\ & & = -\lim _{p\rightarrow \infty }{2}^{2p}\,\frac{\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{ \frac{q} {n+q} }\right )\,\ln (\mathcal{K}_{0})\, \frac{n} {{(n+q)}^{2}} \,d\mu _{K}\right )} {\Omega _{n({2}^{p}-1)}(K)} \\ & & = -n\lim _{p\rightarrow \infty }\frac{\displaystyle\int _{\partial K}\mathcal{K}_{0}^{\frac{{2}^{p}-1} {{2}^{p}} }\,\ln (\mathcal{K}_{0})\,d\mu _{K}} {\Omega _{n({2}^{p}-1)}(K)} \\ & & = -n\,\frac{\displaystyle\int _{\partial K}\mathcal{K}_{0}\,\ln (\mathcal{K}_{0})\,d\mu _{K}} {n\,V ol({K}^{\circ })} =\ln (\Omega _{K}).\end{array}$$

The last equality, due to Eq. (20), completes the proof.

Following from the monotonicity of the sequence (23), we have

Corollary 4.

For any K of class C + 2 containing the origin in its interior, and any integer p ≥ 1,

$$\Omega _{K} \cdot {(nV ol({K}^{\circ }))}^{2} \leq \frac{{(\Omega _{n({2}^{p}-1)}(K))}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p}-2}}, $$
(26)

in particular Ω K ⋅ (nV ol(K ∘ )) 2 ≤ Ω n 2 (K), with equalities everywhere if and only if K is a centered ellipsoid.

Corollary 5.

For any K of class C + 2 containing the origin in its interior, and any integer p ≥ 1,

$$\Omega _{K} \cdot \Omega _{{K}^{\circ }} \leq \frac{{(\Omega _{n({2}^{p}-1)}(K) \cdot \Omega _{n({2}^{p}-1)}({K}^{\circ }))}^{{2}^{p} }} {{({n}^{2}V ol(K) \cdot V ol({K}^{\circ }))}^{{2}^{p} }}, $$
(27)

in particular \(\Omega _{K} \cdot \Omega _{{K}^{\circ }} \leq \frac{\Omega _{n}^{2}(K) \cdot \Omega _{n}^{2}({K}^{\circ })} {{({n}^{2}V ol(K) \cdot V ol({K}^{\circ }))}^{2}}\) , with equalities everywhere if and only if K is a centered ellipsoid in which case the right-hand sides of the two inequalities are equal to 1.

The definition of Ω K can be extended to affine surface areas of negative exponent using a similar result with Proposition 3:

Theorem 3 (Second Alternative Definition of Ω K ). 

For any K of class C + 2 containing the origin in its interior, the sequence

$$\left \{{\left (\frac{\Omega _{-(n+{2}^{p})}({K}^{\circ })} {nV ol(K)} \right )}^{{2}^{p} }\right \}_{p\in \mathbb{N},\ p\geq 1}$$

converges and

$$\lim \limits_{p\rightarrow \infty }{\left (\frac{\Omega _{-(n+{2}^{p})}({K}^{\circ })} {nV ol(K)} \right )}^{{2}^{p} } = \Omega _{K}^{-1}.$$
(28)

Proof.

By applying again Jensen’s inequality for the concave function \(x\mapsto \sqrt{x}\), x > 0, we have, for any integer p ≥ 1,

$$\displaystyle\begin{array}{rcl} \displaystyle\int _{\partial K}\mathcal{K}_{0}^{-\frac{n} {2} }\,d\mu _{K}& \geq & \frac{{\left (\displaystyle\int _{\partial K}\mathcal{K}_{0}^{-\frac{n} {4} }\,d\mu _{K}\right )}^{2}} {n\,V ol(K)} \geq \frac{{\left (\displaystyle\int _{\partial K}\mathcal{K}_{0}^{-\frac{n} {8} }\,d\mu _{K}\right )}^{4}} {{(n\,V ol(K))}^{3}} \\ & \geq & \ldots \geq \frac{{\left (\displaystyle\int _{\partial K}\mathcal{K}_{0}^{-\frac{n} {{2}^{p}} }\,d\mu _{K}\right )}^{{2}^{p} }} {{(n\,V ol(K))}^{{2}^{p}-1}} \geq \ldots \end{array}$$
(29)

therefore the sequence of general term

$$\displaystyle\begin{array}{rcl}{ \left (\frac{\Omega _{-(n+{2}^{p})}({K}^{\circ })} {nV ol(K)} \right )}^{{2}^{p} }& =&{ \left (\frac{\Omega _{-{n}^{2}/(n+{2}^{p})}(K)} {nV ol(K)} \right )}^{{2}^{p} } \\ & =& \left ( \frac{1} {n\,V ol(K)} \cdot \frac{{(\Omega _{-{n}^{2}/(n+{2}^{p})}(K))}^{{2}^{p} }} {{(nV ol(K))}^{{2}^{p}-1}} \right ) \\ \end{array}$$

is monotone. Interchanging K with K  ∘ , we conclude that the sequence \(\left \{{\left (\frac{\Omega _{-(n+{2}^{p})}(K)} {nV ol({K}^{\circ })} \right )}^{{2}^{p} }\right \}_{p\in \mathbb{N},\ p\geq 1}\) is monotone.

We now proceed as in the previous theorem with

$$\displaystyle\begin{array}{rcl} \ln \left [\lim _{p\rightarrow \infty }\frac{{\left (\Omega _{-(n+{2}^{p})}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p} }} \right ]& =& \lim _{p\rightarrow \infty }{2}^{p}\ln \left (\frac{\Omega _{-(n+{2}^{p})}(K)} {nV ol({K}^{\circ })} \right ) \\ & =& -\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dp}\left (\Omega _{-(n+{2}^{p})}(K)\right )} {\Omega _{-(n+{2}^{p})}(K)} \\ & =& -\lim _{p\rightarrow \infty }\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dp}\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{ \frac{n} {{2}^{p}}+1}\right )\,d\mu _{K}\right )} {\Omega _{-(n+{2}^{p})}(K)} \\ & =& n\,\lim _{p\rightarrow \infty }\frac{\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{\frac{n+{2}^{p}} {{2}^{p}} }\right )\,\ln (\mathcal{K}_{0})\,d\mu _{K}\right )} {\Omega _{-(n+{2}^{p})}(K)} \\ & =& n\,\frac{\displaystyle\int _{\partial K}\mathcal{K}_{0}\,\ln (\mathcal{K}_{0})\,d\mu _{K}} {n\,V ol({K}^{\circ })} = -\ln (\Omega _{K}), \\ \end{array}$$

and, using Eq. (20), we complete the proof of the theorem.

While it is known that integrals of the form \(\int _{\partial K}\phi (\mathcal{K}_{0})\,d\mu _{K}\) are SL(n)-invariant, see also [16, 18], considering the results in [30], and others, including for example the next theorem, we conjecture that the set of p-affine surface areas, with algebraic operations, can generate, by taking the closure, all integrals of the above form.

Theorem 4.

For any K of class C + 2 containing the origin in its interior, the SL(n)-invariant \(\Lambda (K) :=\exp \left [ \frac{1} {nV ol(K)}\,\int _{\partial K}\ln (\mathcal{K}_{0})\,d\mu _{K}\right ]\) is the limit, as p → +∞, of the sequence \(\left \{{\left (\frac{\Omega _{-\frac{n} {{2}^{p}} }(K)} {n\,V ol(K)} \right )}^{{2}^{p} }\right \}_{p\in \mathbb{N},\ p>1}.\)

Proof.

The claim follows directly from

$$\displaystyle\begin{array}{lll} {} & {} & \ln \left [\lim _{p\rightarrow \infty }\frac{{\left (\Omega _{-n/{2}^{p}}(K)\right )}^{{2}^{p} }} {{(nV ol({K}^{\circ }))}^{{2}^{p} }} \right ] \\ & & =\lim _{p\rightarrow \infty }{2}^{p}\ln \left (\frac{\Omega _{-n/{2}^{p}}(K)} {nV ol({K}^{\circ })} \right ) \\ & & = -\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dp}\left (\Omega _{-n/{2}^{p}}(K)\right )} {\Omega _{-n/{2}^{p}}(K)} \\ & & = -\lim _{p\rightarrow \infty }\frac{{2}^{p}} {\ln 2} \,\frac{ \frac{d} {dp}\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{- \frac{1} {{2}^{p}-1} }\right )\,d\mu _{K}\right )} {\Omega _{-n/{2}^{p}}(K)} \\ & & =\lim _{p\rightarrow \infty } \frac{{2}^{2p}} {{({2}^{p} - 1)}^{2}}\,\frac{\left (\displaystyle\int _{\partial K}\exp \left (\ln \mathcal{K}_{0}^{- \frac{1} {{2}^{p}-1} }\right )\,\ln (\mathcal{K}_{0})\,d\mu _{K}\right )} {\Omega _{-n/{2}^{p}}(K)} \\ & & =\lim _{p\rightarrow \infty } \frac{{2}^{2p}} {{({2}^{p} - 1)}^{2}}\,\frac{\left (\displaystyle\int _{\partial K}\mathcal{K}_{0}^{- \frac{1} {{2}^{p}-1} }\,\ln (\mathcal{K}_{0})\,d\mu _{K}\right )} {\Omega _{-n/{2}^{p}}(K)} \\ & & = \frac{\displaystyle\int _{\partial K}\ln (\mathcal{K}_{0})\,d\mu _{K}} {n\,V ol(K)} =\ln (\Lambda _{K}).\end{array}$$