Extensions of translation invariant valuations on polytopes

TL;DR

This paper investigates how translation invariant valuations on convex polytopes can be extended to all convex bodies, introducing flag support measures to facilitate this extension and analyzing their properties.

Contribution

It introduces flag support measures for convex bodies and explores the continuity conditions needed for extending valuations from polytopes to all convex bodies.

Findings

Flag support measures are introduced and characterized.

Certain continuity properties are identified as sufficient for extension.

The properties of these measures are analyzed in detail.

Abstract

We study translation invariant, real-valued valuations on the class of convex polytopes in Euclidean space and discuss which continuity properties are sufficient for an extension of such valuations to all convex bodies. For this purpose, we introduce flag support measures of convex bodies via a local Steiner formula and derive some of the properties of these measures.