Brunn-Minkowski Inequality for θ-Convolution Bodies via Ball's Bodies

Abstract

We consider the problem of finding the best function ${\phi }_n:[0,1]\to R$ such that for any pair of convex bodies $K,L\in R^n$ the following Brunn-Minkowski type inequality holds $\begin{array}{c}|K{+}_{\theta }{L|}^{\frac{1}{n}}\ge {\phi }_n{\left(\theta \right)\left(\right|K|}^{\frac{1}{n}}+{\left|L\right|}^{\frac{1}{n}}),\end{array}$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 ${\left(\frac{3}{4}\right)}^n\le \theta \le 1$, characterizing the equality cases.

Keywords: Ball‘s bodies; Brunn–Minkowski inequality; $\theta$-Convolution bodies.