Improved $L_p-$mixed volume inequality for convex bodies

TL;DR

This paper establishes a sharp quantitative version of the $L_p$-mixed volume inequality for convex bodies by leveraging an improved Jensen inequality, which also leads to a refined $L_p$-Brunn-Minkowski inequality.

Contribution

It introduces a new sharp quantitative $L_p$-mixed volume inequality using an improved Jensen inequality, generalizing known entropy inequalities.

Findings

Proves a sharp quantitative $L_p$-mixed volume inequality.

Derives a sharp quantitative $L_p$-Brunn-Minkowski inequality as a corollary.

Utilizes an improved Jensen inequality related to Tsallis entropy.

Abstract

A sharp quantitative version of the $L_p-$mixed volume inequality is established. This is achieved by exploiting an improved Jensen inequality. This inequality is a generalization of Pinsker-Csisz\'ar-Kullback inequality for the Tsallis entropy. Finally, a sharp quantitative version of the $L_p-$Brunn-Minkowski inequality is also proved as a corollary.