Groups of convex bodies

Richard Hepworth* (Corresponding Author)

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper we introduce and study a topological abelian group of convex bodies, analogous to the scissors congruence group and McMullen’s polytope algebra, with the universal property that continuous valuations on convex bodies correspond to continuous homomorphisms on the group of convex bodies. To study this group, we first obtain a version of McMullen polynomiality for valuations that take values not in fields or vector spaces, but in abelian groups. Using this, we are able to equip the group of convex bodies with a grading that consists of real vector spaces in all positive degrees, mirroring one of the main structural properties of the polytope algebra. It is hoped that this work can serve as the starting point for a K-theoretic interpretation of valuations on convex bodies.
Original languageEnglish
Article number217
Number of pages17
JournalGeometriae Dedicata
Early online date27 Jun 2023
Publication statusPublished - 27 Jun 2023

Bibliographical note

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  • Convex bodies
  • McMullen polynomiality
  • K-theory


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