The Hadwiger theorem on convex functions, IV: The Klain approach

TL;DR

This paper presents new proofs of the Hadwiger theorem for convex functions, establishes the Klain-Schneider theorem in this context, and introduces an extension theorem linked to the Abel transform, advancing valuation theory.

Contribution

It provides novel proofs of key theorems for convex functions and extends valuation theory through an innovative extension theorem connected to the Abel transform.

Findings

New proofs of Hadwiger theorem for smooth and continuous valuations

Establishment of Klain-Schneider theorem for convex functions

Extension theorem for valuations on lower-dimensional domains

Abstract

New proofs of the Hadwiger theorem for smooth and for continuous valuations on convex functions are obtained, and the Klain-Schneider theorem on convex functions is established. In addition, an extension theorem for valuations defined on functions with lower dimensional domains is proved, and its connection to the Abel transform is explained.