From the Brunn-Minkowski inequality to a class of generalized Poincar\'{e}-type inequalities for torsional rigidity

TL;DR

This paper derives a new class of Poincaré-type inequalities for torsional rigidity on convex bodies, utilizing the concavity properties of the Brunn-Minkowski inequality, advancing geometric analysis techniques.

Contribution

It introduces generalized Poincaré inequalities for torsional rigidity based on the concavity of the Brunn-Minkowski inequality for convex bodies.

Findings

Established a new class of inequalities for torsional rigidity

Linked Brunn-Minkowski concavity to boundary inequalities

Extended geometric analysis methods for convex bodies

Abstract

In this paper, we establish a class of generalized Poincar\'{e}-type inequalities for torsional rigidity on the boundary of a convex body of class $C^{2}_{+}$ in $\rnnn$ by using the concavity of related Brunn-Minkowski inequality.