The Orlicz-Petty bodies

Abstract

This paper is dedicated to the Orlicz-Petty bodies. We first propose the homogeneous Orlicz affine and geominimal surface areas, and establish their basic properties such as homogeneity, affine invariance and affine isoperimetric inequalities. We also prove that the homogeneous geominimal surface areas are continuous, under certain conditions, on the set of convex bodies in terms of the Hausdorff distance. Our proofs rely on the existence of the Orlicz-Petty bodies and the uniform boundedness of the Orlicz-Petty bodies of a convergent sequence of convex bodies. Similar results for the nonhomogeneous Orlicz geominimal surface areas are proved as well.

On the general dual Orlicz-Minkowski problem

“…inf / sup Note that closely related to (1.3) are the Orlicz affine and geominimal surface areas, which were proposed in [57,58,62]. In fact, one can observe that (1.2) not only generalizes (1.3), but also is "dual" to (1.3).…”

“…The classical geominimal surface area [46,47] and its L p or Orlicz extensions (see e.g., [39,56,57,58,62]) are central objects in convex geometry. When studying the properties of various geominimal surface areas, the Petty body or its generalizations play fundamental roles.…”

“…When studying the properties of various geominimal surface areas, the Petty body or its generalizations play fundamental roles. In short, the Orlicz-Petty bodies are the solutions to the following optimization problems [58,62]: where ϕ ∈ I , and V ϕ (K, L) and V ϕ (K, L) are the Orlicz L ϕ mixed volumes of K, L ∈ K n (o) defined by (see e.g., [12,54,62]):…”

“…The L p Minkowski problem and the L p affine surface area were apparently developed in completely different approaches, however, they were nicely connected through the L p geominimal surface area and the L p Petty bodies [39,56,62]. As the bridge to connect several geometries (affine, Minkowski and relative), the L p geominimal surface area is crucial in convex geometry and, in particular, share many properties similar to those for the L p affine surface area.…”

“…where B n is the unit Euclidean ball in R n , V (·) stands for the volume, L • denotes the polar body of L ∈ K n (o) , and h L is the support function of L (see Section 2 for notations). As explained in [34], the L p Minkowski problem can be viewed as the "polarity" of (1.1) (in particular, for µ nice enough such as µ being even) aiming to find convex bodies (ideally in K n (o) ) to solve the optimization problem similar to (1.1), namely with L • replaced by L. On the other hand, the L p affine surface area of K ∈ K n (o) can be defined through a formula similar to (1.1) for µ being the L p surface area measure of K, but with L ∈ K n (o) and h L • replaced by L belong to star bodies about the origin and, respectively, ρ −1 L where ρ L is the radial function of L (see [39,56,62] for more details). The main purpose of this article is to give a systematic study to the general dual-polar Orlicz-Minkowski problem, which extends problem (1.1) in the arguably most general way: with the function t p (from the integrand of the objective functional) and V (L) in problem (1.1) replaced by a (general nonhomogeneous) continuous function ϕ : (0, ∞) → (0, ∞) and, respectively, V G (L), the general dual volume of L, formulated by V G (L) = S n−1 G(ρ L (u), u) du with du the spherical measure of S n−1 .…”