Hadwiger’s Theorem for definable functions ☆

Abstract Hadwiger’s Theorem states that E n -invariant convex-continuous valuations of definable sets in R n are linear combinations of intrinsic volumes. We lift this result from sets to data distributions over sets, specifically, to definable R -valued functions on R n . 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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