Hadwiger's Theorem for Definable Functions

TL;DR

This paper extends Hadwiger's Theorem from convex sets to definable functions, establishing a classification of invariant valuations on functions that generalize intrinsic volumes.

Contribution

It generalizes Hadwiger's Theorem from sets to definable functions, introducing a dual pair of classification theorems for valuations on functions.

Findings

Established a dual pair of Hadwiger classification theorems for functions

Generalized intrinsic volumes to non-linear valuations on functions

Provided a framework for Euclidean-invariant valuations on definable functions

Abstract

Hadwiger's Theorem states that Euclidean-invariant convex-continuous valuations of definable sets are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable real-valued functions on n-dimensional Euclidean space. This generalizes intrinsic volumes to (dual pairs) of non-linear valuations on functions and provides a dual pair of Hadwiger classification theorems.