Generalized translation invariant valuations and the polytope algebra

Andreas Bernig, Dmitry Faifman

Research output: Contribution to journalArticlepeer-review

Abstract

We study the space of generalized translation invariant valuations on a finite-dimensional vector space and construct a partial convolution which extends the convolution of smooth translation invariant valuations. Our main theorem is that McMullen's polytope algebra is a subalgebra of the (partial) convolution algebra of generalized translation invariant valuations. More precisely, we show that the polytope algebra embeds injectively into the space of generalized translation invariant valuations and that for polytopes in general position, the convolution is defined and corresponds to the product in the polytope algebra.

Original languageEnglish
Pages (from-to)36-72
Number of pages37
JournalAdvances in Mathematics
Volume290
DOIs
StatePublished - 26 Feb 2016
Externally publishedYes

Keywords

  • Convex geometry
  • Integral geometry
  • Polytopes
  • Valuations

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