Groups of convex bodies

TL;DR

This paper introduces a topological abelian group of convex bodies that generalizes existing algebraic structures and links valuations to homomorphisms, aiming to develop a K-theoretic framework for convex geometry.

Contribution

It constructs a new topological abelian group of convex bodies with a grading structure, extending McMullen polynomiality to group-valued valuations, and proposes a foundation for K-theoretic interpretation.

Findings

Established a universal property linking valuations and homomorphisms.

Developed a grading of the convex bodies group with real vector space components.

Extended McMullen polynomiality to group-valued valuations.

Abstract

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.