Orlicz Mixed Affine Quermassintegrals

TL;DR

This paper introduces Orlicz mixed affine quermassintegrals, generalizing classical affine quermassintegrals within the Orlicz-Brunn-Minkowski framework, and establishes related inequalities extending Lutwak's results.

Contribution

It develops the concept of Orlicz mixed affine quermassintegrals and proves new inequalities, broadening the scope of affine geometry in Orlicz spaces.

Findings

Proves an Orlicz-Minkowski inequality for the new integrals.

Establishes an Orlicz-Brunn-Minkowski inequality as a generalization.

Extends Lutwak's classical inequalities to the Orlicz setting.

Abstract

Lutwak's notion of affine quermassintegrals of a convex body quickly became of great importance in convex and affine geometry and more recently, also in asymptotic geometric analysis. In this note we introduce the notion of Orlicz mixed affine quermassintegrals of a convex body as a generalization of the affine quermassintegrals in the framework of the Orlicz-Brunn-Minkowski theory. We prove a Orlicz-Minkowski inequality for the Orlicz mixed and affine quermassintegrals, and an Orlicz-Brunn-Minkowski inequality, which provides of a direct generalization of Lutwak's Brunn-Minkowski inequality for affine quermassintegrals, in Orlicz spaces.