The Hadwiger theorem on convex functions, III: Steiner formulas and mixed Monge-Amp\`ere measures

TL;DR

This paper develops a comprehensive family of Steiner formulas for convex functions, providing explicit representations of functional intrinsic volumes via mixed Monge-Ampère measures and extending Hadwiger's theorem to convex functions.

Contribution

It introduces a complete family of functional Steiner formulas and a new version of Hadwiger's theorem for convex functions, linking intrinsic volumes with Monge-Ampère measures.

Findings

Explicit representation of functional intrinsic volumes.

A new version of Hadwiger's theorem for convex functions.

Establishment of a complete family of functional Steiner formulas.

Abstract

A complete family of functional Steiner formulas is established. As applications, an explicit representation of functional intrinsic volumes using special mixed Monge-Amp\`ere measures and a new version of the Hadwiger theorem on convex functions are obtained.