Hadwiger's Theorem for Definable Functions

2012/03/28 by Baryshnikov, Yuliy, Ghrist, Robert, Wright, Matthew
#49Q15 (Primary) 53C65 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1203.6120

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.

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