Curvature functionals on convex bodies

TL;DR

This paper explores weighted $L_p$ affine surface areas in convex geometry, establishing their valuation property, isoperimetric inequalities, and connections to information divergences like Kullback-Leibler and Rényi.

Contribution

It introduces valuation properties and isoperimetric inequalities for weighted $L_p$ affine surface areas, linking them to information divergence measures.

Findings

Weighted $L_p$ affine surface areas are valuations.

Established isoperimetric inequalities for these surface areas.

Connected these geometric measures to Kullback-Leibler and Rényi divergences.

Abstract

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to $f$ divergences of the cone measures of the convex body and its polar, namely the Kullback-Leibler divergence and the R\'enyi-divergence.