Inequalities and counterexamples for functional intrinsic volumes and beyond

TL;DR

This paper investigates the limitations of analytic inequalities for functional intrinsic volumes, provides counterexamples, and introduces new variational functionals with established inequalities, extending classical geometric results.

Contribution

It demonstrates the failure of certain inequalities for functional intrinsic volumes and introduces a new family of variational functionals with proven inequalities, extending classical geometric inequalities.

Findings

Counterexamples show failure of Brunn-Minkowski-type inequalities for functional intrinsic volumes.

Introduction of a family of variational functionals satisfying Wulff-type inequalities.

Generalization of Alexandrov-Fenchel-type inequalities and recovery of Pólya-Szegő-type inequalities.

Abstract

We show that analytic analogs of Brunn-Minkowski-type inequalities fail for functional intrinsic volumes on convex functions. This is demonstrated both through counterexamples and by connecting the problem to results of Colesanti, Hug, and Saor\'in G\'omez. By restricting to a smaller set of admissible functions, we then introduce a family of variational functionals and establish Wulff-type inequalities for these quantities. In addition, we derive inequalities for the corresponding family of mixed functionals, thereby generalizing an earlier Alexandrov-Fenchel-type inequality by Klartag and recovering a special case of a recent P\'olya-Szeg\H{o}-type inequality by Bianchi, Cianchi, and Gronchi.