Brunn-Minkowski inequality for $\theta$-convolution bodies via Ball's bodies

TL;DR

This paper establishes a sharp Brunn-Minkowski type inequality for $ heta$-convolution bodies of convex sets, identifying the optimal function and equality cases using Ball's bodies of $ heta$-concave functions.

Contribution

It determines the best function $ heta o rac{1}{n}$-power inequality for $ heta$-convolution bodies, extending classical Brunn-Minkowski results with sharp bounds and equality characterization.

Findings

Derived the optimal function $ heta o rac{1}{n}$-power inequality.

Proved a sharp inclusion of Ball's bodies in super-level sets of $ heta$-concave functions.

Characterized the cases of equality in the inequality.

Abstract

We consider the problem of finding the best function $\varphi_n:[0,1]\to\mathbb{R}$ such that for any pair of convex bodies $K,L\in\mathbb{R}^n$ the following Brunn-Minkowski type inequality holds $$ |K+_\theta L|^\frac{1}{n}\geq\varphi_n(\theta)(|K|^\frac{1}{n}+|L|^\frac{1}{n}), $$ where $K+_\theta L$ is the $\theta$-convolution body of $K$ and $L$. We prove a sharp inclusion of the family of Ball's bodies of an $\alpha$-concave function in its super-level sets in order to provide the best possible function in the range $\left(\frac{3}{4}\right)^n\leq\theta\leq1$, characterizing the equality cases.