Extremal general affine surface areas

Abstract

For a convex body K in ${\mathbb{R}}^n$, we introduce and study the extremal general affine surface areas, defined by

$${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }\subset K}{\sup }{\mathrm{as}}_{\phi }(K),{\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }\supset K}{\inf }{\mathrm{as}}_{\psi }(K)$$

where ${\mathrm{as}}_{\phi }(K)$ and ${\mathrm{as}}_{\psi }(K)$ are the $L_{\phi }$ and $L_{\psi }$ affine surface area of K, respectively. We prove that there exist extremal convex bodies that achieve the supremum and infimum, and that the functionals ${\mathrm{\text{IS}}}_{\phi }$ and ${\mathrm{\text{os}}}_{\psi }$ are continuous. In our main results, we prove Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas. This article may be regarded as an Orlicz extension of the recent work of Giladi, Huang, Schütt and Werner (2020), who introduced and studied the extremal $L_p$ affine surface areas.

Introduction

The celebrated result of Fritz John [27] says that every convex body K in ${\mathbb{R}}^n$ contains a unique ellipsoid of maximum volume, now called the John ellipsoid of K. The minimum volume ellipsoid containing K is also unique and is called the Löwner ellipsoid of K. The John and Löwner ellipsoids are related by polar duality: If K contains the origin in its interior, then the John ellipsoid is the polar body of the Löwner ellipsoid, and vice versa. These ellipsoids are cornerstones of asymptotic convex geometry, arising in concentration of volume, the reverse isoperimetric inequality, the hyperplane conjecture and many more [1], [11]. Other prominent applications of the John and Löwner ellipsoids can be found in Banach space geometry [10], [15], [44], [55], extremal problems [2], [7], [17], [48], polytopal approximation of convex bodies [20], [33], [52], linear programming [19], [56] and statistics [14], [45], [58].

In this article, we study an analogue of John's theorem when volume is replaced by the general $L_{\phi }$ or $L_{\psi }$ affine surface area, which is defined in terms of a special type of concave function φ or convex function ψ, respectively. More specifically, instead of considering the ellipsoid of maximum (resp. minimum) volume contained in (containing) K, we will consider the convex body of maximum $L_{\phi }$ (minimum $L_{\psi }$) affine surface contained in (containing) K. For a convex body K in ${\mathbb{R}}^n$ with centroid at the origin, we define the inner maximal $L_{\mathit{\phi }}$ affine surface area ${\mathrm{\text{IS}}}_{\phi }(K)$ and outer minimal $L_{\mathit{\psi }}$ affine surface area ${\mathrm{\text{os}}}_{\psi }(K)$ of K by

$${\mathrm{\text{IS}}}_{\phi }(K):=\underset{K^{\prime }\subset K}{\sup }{\mathrm{as}}_{\phi }(K^{\prime })$$

$${\mathrm{\text{os}}}_{\psi }(K):=\underset{K^{\prime }\supset K}{\inf }{\mathrm{as}}_{\psi }(K^{\prime }).$$

Here ${\mathrm{as}}_{\phi }(K)$ and ${\mathrm{as}}_{\psi }(K)$ denote the general $L_{\phi }$ and $L_{\psi }$ affine surface area of K, respectively, which are extensions of the classical affine surface area defined by Schütt and Werner [51]. The supremum and infimum are taken over all $K^{\prime }$ with centroid at the origin (see Section 2.1 for the definitions).

The special case of (1) when ${\mathrm{as}}_{\phi }$ is replaced by the classical affine surface area has attracted considerable interest. Inscribed bodies of maximal affine surface area have found applications to discrete geometry [3], [4], [5], [6], geometric probability [5] and variational problems in differential equations [32], [57]. The existence of a convex body $K_{\max }\subset K$ of maximum affine surface area follows from the upper semicontinuity of the affine surface area [4], [50]. The regularity properties of $K_{\max }$ were studied by Sheng, Trudinger and Wang [54]. For planar convex bodies, Bárány proved that $K_{\max }$ is unique, and showed a remarkable relationship between $K_{\max }$ and the limit shape of the convex hull of certain lattice points [4], [6] or random points [5] contained in K. Bárány and Prodromou [6] showed that if K is a planar convex body, then $K_{\max }$ is of elliptic type, and if K is of elliptic type then $K_{\max }=K$. Schneider [50] used a new method to extend the latter result to all dimensions, showing that if K is a convex body in ${\mathbb{R}}^n$ of elliptic type, then $K_{\max }=K$.

Recently, Giladi, Huang, Schütt and Werner [18] introduced and studied the extremal $L_p$ affine surface areas

$${\mathrm{\text{IS}}}_p(K):=\underset{K^{\prime }\subset K}{\sup }{\mathrm{as}}_p(K^{\prime })$$

$${\mathrm{\text{os}}}_p(K):=\underset{K^{\prime }\supset K}{\inf }{\mathrm{as}}_p(K^{\prime })$$

where ${\mathrm{as}}_p(K)$ is the $L_p$ affine surface area of K. It was shown in [18] that for any convex body K with centroid at the origin, the extremal bodies exist, and continuity and affine isoperimetric inequalities were proved for the functionals ${\mathrm{\text{IS}}}_p$ and ${\mathrm{\text{os}}}_p$. Asymptotic estimates were also given for ${\mathrm{\text{IS}}}_p(K)$ and ${\mathrm{\text{os}}}_p(K)$ in terms of powers of the volume of K using the Löwner ellipsoid and thin shell estimate of [21].

When a certain concave or convex function depending on n and p is chosen in (1) or (2), respectively, we recover (3) and (4) as special cases. Thus, the new definitions (1) and (2) may be thought of as Orlicz extensions of the definitions (3) and (4) (see Section 6.4 below for the details). A key difference between the $L_p$ affine surface areas and the general $L_{\phi }$ and $L_{\psi }$ affine surface areas is that the former are homogeneous, while the latter, in general, are not. For our purposes, the lack of homogeneity will only present difficulties for the $L_{\phi }$ affine surface areas. To surmount this obstacle, we will restrict our attention to certain subclasses of concave functions φ satisfying mild growth rate conditions. These subclasses contain many functions that are not homogeneous of any degree, which distinguishes the results in this work from those in [18]. At the same time, these classes contain the homogeneous functions considered in the $L_p$ setting of [18].

In the present paper, we prove existence and continuity of the extremal general affine surface areas (1) and (2). Our main result is a Blaschke-Santaló type inequality for the inner maximal $L_{\phi }$ affine surface area. More specifically, in Theorem 6.4 we show that for any convex body K in ${\mathbb{R}}^n$ with centroid at the origin and any concave function φ satisfying some prescribed conditions,

$${\mathrm{\text{IS}}}_{\phi }(K){\mathrm{\text{IS}}}_{\phi }(K^{\circ })\le {\mathrm{\text{IS}}}_{\phi }{(B_n)}^2$$

with equality if and only if K is an ellipsoid (here $B_n$ denotes the Euclidean unit ball in ${\mathbb{R}}^n$ centered at the origin). Finally, we prove an inverse Santaló type inequality for the outer minimal $L_{\psi }$ affine surface area.

In Section 2, we state definitions and notation used throughout the paper, and provide the relevant background on the general $L_{\phi }$ and $L_{\psi }$ affine surface areas. In Section 3, we use a polar duality relation of Ludwig [34] for the $L_{\phi }$ affine surface area to define the subclasses of concave functions φ we will consider. Next, in Section 4 we define the extremal general affine surface areas and state their existence, monotonicity and continuity properties; for the reader's convenience, the proofs of these properties are included at the end of the paper in Section 7. The lemmas that will be used in the proofs of the main results are in Section 5, and in Section 6 we prove affine isoperimetric inequalities for the extremal general affine surface areas. In Subsections 6.2 and 6.3 we prove our main results, which are Blaschke-Santaló type inequalities and inverse Santaló type inequalities for the extremal general affine surface areas.