Hadwiger's Theorem for definable functions

Y. Baryshnikov, R. Ghrist, M. Wright

Research output: Contribution to journalArticlepeer-review

24 Scopus citations


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

Original languageEnglish (US)
Pages (from-to)573-586
Number of pages14
JournalAdvances in Mathematics
StatePublished - Oct 1 2013

Bibliographical note

Funding Information:
This work was supported by DARPA # HR0011-07-1-0002 and ONR N000140810668 .


  • Euler characteristic
  • Hadwiger measure
  • Intrinsic volumes
  • Valuations


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