Orlicz--Lorentz centroid bodies

TL;DR

This paper introduces a new class of centroid bodies called Orlicz--Lorentz centroid bodies, extending existing concepts, and proves a sharp affine isoperimetric inequality relating their volume to the original convex body.

Contribution

It extends the centroid body operator to Orlicz--Lorentz settings and establishes a sharp affine isoperimetric inequality for these bodies.

Findings

Established the Orlicz--Lorentz centroid body operator.

Proved a sharp affine isoperimetric inequality for these bodies.

Bounded the volume of the centroid body from below by the original convex body's volume.

Abstract

We extend the definition of the centroid body operator to an Orlicz--Lorentz centroid body operator on the star bodies in $\mathbb R^n$, and establish the sharp affine isoperimetric inequality that bounds (from below) the volume of the Orlicz--Lorentz centroid body of any convex body containing the origin in its interior by the volume of this convex body.