Sufficient and Necessary Condition of the Lp-Brunn-Minkowski Inequality Conjecture for p\in[0,1)

TL;DR

This paper explores the conditions under which the Lp-Brunn-Minkowski inequality holds for p in [0,1), linking the conjecture to eigenvalue bounds of Aleksandrov's problem, thus advancing understanding of convex geometric inequalities.

Contribution

It establishes a simple equivalence between the conjecture's validity and a lower bound on the third eigenvalue of Aleksandrov's problem.

Findings

Equivalence between the conjecture and eigenvalue bounds clarified.

Provides a new perspective linking geometric inequalities to spectral properties.

Simplifies the understanding of the Lp-Brunn-Minkowski conjecture for p in [0,1).

Abstract

The Lp-Brunn-Minkowski inequality palys a central role in the Brunn-Minkowski theory proposed by Firey [13] in 60's and developed by Lutwak [26,27] in 90's, which generalizes the classical Brunn-Minkowski inequality by Lp-sum of convex bodies. The inequality has been established by Firey for p>1 and later been conjectured by Borozky-Lutwak-Yang-Zhang [5] for p\in[0,1). (see also [23,7]) The validity of this conjecture was verified for the planar case in [5], and for the higher dimensional case when p closing to 1 by Chen-Huang-Li-Liu [7]. (see alsoo a local version by Kolesnikov-Milman [23]) In this short note, we give a simple argument clarifying the equivalence between the full conjecture and a lower bound of the third eigenvalue of the Aleksandrov's problem.