The Hadwiger theorem on convex functions, II: Cauchy-Kubota formulas

TL;DR

This paper extends the Hadwiger theorem to convex functions, providing explicit formulas for functional intrinsic volumes and exploring their relation to classical valuations.

Contribution

It introduces a new version of the Hadwiger theorem for convex functions and derives functional Cauchy-Kubota formulas for explicit volume representations.

Findings

Explicit representation of functional intrinsic volumes.

Connections established between functional and classical intrinsic volumes.

Classification of non-negative valuations.

Abstract

A new version of the Hadwiger theorem on convex functions is established and an explicit representation of functional intrinsic volumes is found using new functional Cauchy-Kubota formulas. In addition, connections between functional intrinsic volumes and their classical counterparts are obtained and non-negative valuations are classified.