Non-continuous valuations on convex bodies and a new characterization of volume

TL;DR

This paper uses automatic continuity techniques to strengthen the classical characterization of volume on convex bodies, revealing new insights into the degrees of homogeneity of valuations.

Contribution

It introduces an automatic continuity theorem for valuations on parallelotopes and provides a new characterization of volume, identifying possible degrees of homogeneity.

Findings

Strengthened classical volume characterization

Identified degrees of homogeneity as [0,n-1]∪{n}

Established automatic continuity for valuations on parallelotopes

Abstract

This paper investigates the use of automatic continuity techniques in the context of valuations on convex bodies. We first provide an automatic continuity theorem for valuations restricted to parallelotopes with respect to a fixed basis. This result in combination with a counting argument provides a strengthened version of a classical characterization of volume due to Hadwiger. As a byproduct of the proof it is shown that $[0,n-1]\cup\{n\}$ are precisely the possible degrees of homogeneity of bounded translation invariant valuations on $n$-dimensional convex bodies.