Weighted Brunn-Minkowski Theory I: On Weighted Surface Area Measures

TL;DR

This paper extends the classical Brunn-Minkowski theory to weighted measures in convex geometry, introducing new formulas and inequalities that apply to various measures, including the Gaussian, with potential broad implications.

Contribution

It develops weighted versions of surface area measures and mixed volumes, providing new integral formulas and inequalities in the context of Borel measures with density.

Findings

Derived a new integral formula for the mixed measure of three convex bodies.

Proved a Bézout-type inequality for rotationally invariant log-concave measures.

Results are applicable even for the standard Gaussian measure.

Abstract

The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in $\mathbb{R}^n$-- in particular, the weighted versions of mixed volumes (the so-called mixed measures) when dealing with up to three distinct convex bodies. We then formulate and analyze weighted versions of classical surface area measures, and obtain a new integral formula for the mixed measure of three bodies. As an application, we prove a B\'ezout-type inequality for rotational invariant log-concave measures, generalizing a result by Artstein-Avidan, Florentin and Ostrover. The results are new and interesting even for the special case of the standard Gaussian measure.