The log-Brunn-Minkowski inequality in $\mathbb{R}^3$

TL;DR

This paper extends the log-Brunn-Minkowski inequality and related inequalities from the plane to three-dimensional space, providing new theoretical results for convex bodies in D.

Contribution

It establishes the log-Brunn-Minkowski, log-Minkowski, and $L_p$-Minkowski inequalities for convex bodies in D, advancing the understanding of these inequalities beyond the plane.

Findings

Proved the log-Brunn-Minkowski inequality in D.

Extended the log-Minkowski and $L_p$-Minkowski inequalities to three dimensions.

Established the $L_p$-Brunn-Minkowski inequality in D.

Abstract

B\"or\"oczky, Lutwak, Yang and Zhang recently proved the log-Brunn-Minkowski inequality which is stronger than the classical Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane. This paper establishes the log-Brunn-Minkowski, log-Minkowski, $L_p$-Minkowski and $L_p$-Brunn-Minkowski inequalities for two convex bodies in $\mathbb{R}^3$.