On Discrete LOG-Brunn--Minkowski Type Inequalities

Abstract

The conjectured log-Brunn--Minkowski inequality says that the volume of centrally symmetric convex bodies $K,L\subset\mathbb{R}^n$ satisfies ${vol}\bigl((1-\lambda)\cdot K+_0\lambda\cdot L\bigr) \geq{vol}(K)^{1-\lambda}{vol}(L)^\lambda$, $\lambda\in(0,1)$, and is known to be true in the plane and for particular classes of symmetric convex bodies in $\mathbb{R}^n$. In this paper, we get some discrete log-Brunn--Minkowski type inequalities for the lattice point enumerator. Among others, we show that if $K,L\subset\mathbb{R}^n$ are unconditional convex bodies and $\lambda\in(0,1)$, then ${G}_n((1-\lambda)\cdot(K+C_n)+_0\lambda\cdot(L+C_n)+(-\frac{1}{2},\frac{1}{2})^n) \geq{G}_n(K)^{1-\lambda}{G}_n(L)^\lambda, $ where $C_n=[-1/2,1/2]^n$. Neither $C_n$ nor $(-1/2,1/2)^n$ can be removed. Furthermore, it implies the (volume) log-Brunn--Minkowski inequality for unconditional convex bodies. The corresponding results in the $L_p$ setting for $0<p<1$ are also obtained.