The Orlicz-Petty bodies

TL;DR

This paper introduces and studies the properties of Orlicz-Petty bodies, including affine surface areas and isoperimetric inequalities, expanding the understanding of convex geometric analysis.

Contribution

It proposes homogeneous Orlicz affine and geominimal surface areas and establishes their fundamental properties and continuity, based on the existence of Orlicz-Petty bodies.

Findings

Homogeneous Orlicz affine surface areas are affine invariant.

Homogeneous geominimal surface areas are continuous under Hausdorff convergence.

Results extend to nonhomogeneous Orlicz geominimal surface areas.

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.