Brunn-Minkowski and Zhang inequalities for Convolution Bodies

TL;DR

This paper introduces a quantitative extension of Minkowski sum and $ heta$-convolution for convex bodies, leading to new versions of Brunn-Minkowski and Zhang inequalities and exploring properties of convolution and polar projection bodies.

Contribution

It develops a generalized $ heta$-convolution framework for multiple convex bodies, extending classical inequalities and properties in convex geometry.

Findings

Extended Brunn-Minkowski and Zhang inequalities for convolution bodies

Introduced a multi-set $ heta$-convolution extension

Derived new properties involving convolution and polar projection bodies

Abstract

A quantitative version of Minkowski sum, extending the definition of $\theta$-convolution of convex bodies, is studied to obtain extensions of the Brunn-Minkowski and Zhang inequalities, as well as, other interesting properties on Convex Geometry involving convolution bodies or polar projection bodies. The extension of this new version to more than two sets is also given.