Affine isoperimetric inequalities in the functional Orlicz-Brunn-Minkowski theory

TL;DR

This paper introduces and develops the theory of Orlicz affine and geominimal surface areas for convex and s-concave functions, establishing fundamental properties and inequalities in the functional Orlicz-Brunn-Minkowski framework.

Contribution

It presents the first systematic development of Orlicz affine invariants for functions, including properties and isoperimetric inequalities, expanding the classical geometric theory to a functional setting.

Findings

Established basic properties of the new functional affine invariants.

Proved functional affine isoperimetric inequalities.

Derived functional Santaló type inequalities.

Abstract

In this paper, we develop a basic theory of Orlicz affine and geominimal surface areas for convex and $s$-concave functions. We prove some basic properties for these newly introduced functional affine invariants and establish related functional affine isoperimetric inequalities as well as functional Santal\'o type inequalities.