From the Brunn-Minkowski inequality to a class of Poincar\'e type inequalities

TL;DR

This paper derives a class of Poincaré-type inequalities on convex bodies with smooth boundaries and positive Gauss curvature, using the Brunn-Minkowski inequality as a foundational tool.

Contribution

It introduces a novel method linking the Brunn-Minkowski inequality to boundary Poincaré inequalities for convex bodies.

Findings

Establishes a new connection between geometric inequalities and boundary analysis.

Provides a framework for deriving Poincaré inequalities from classical convex geometry results.

Abstract

We present an argument which leads from the Brunn-Minkowski inequality to a Poincare' type inequality on the boundary of convex bodies with smooth boundary and positive Gauss curvature