$L_p$-Brunn-Minkowski inequality for $p\in (1-\frac{c}{n^{\frac{3}{2}}}, 1)$

TL;DR

This paper proves a global $L_p$-Brunn-Minkowski inequality for certain $p$ values near 1, extending local results and establishing uniqueness for related Minkowski problems with specific measure conditions.

Contribution

It extends local $L_p$-Brunn-Minkowski inequalities to global ones and provides new uniqueness results for the $L_p$-Minkowski and Logarithmic Minkowski problems.

Findings

Established the global $L_p$-Brunn-Minkowski inequality for $p$ near 1.

Proved uniqueness for the $L_p$-Minkowski problem with even positive $C^{eta}$ densities.

Proved uniqueness for the Logarithmic Minkowski problem with small perturbations of uniform density.

Abstract

Kolesnikov-Milman [9] established a local $L_p$-Brunn-Minkowski inequality for $p\in(1-c/n^{\frac{3}{2}},1).$ Based on their local uniqueness results for the $L_p$-Minkowski problem, we prove in this paper the (global) $L_p$-Brunn-Minkowski inequality. Two uniqueness results are also obtained: the first one is for the $L_p$-Minkowski problem when $p\in (1-c/n^{\frac{3}{2}}, 1)$ for general measure with even positive $C^{\alpha}$ density, and the second one is for the Logarithmic Minkowski problem when the density of measure is a small $C^{\alpha}$ even perturbation of the uniform density.