On extendability by continuity of valuations on convex polytopes

TL;DR

This paper investigates conditions under which valuations defined on convex polytopes can be extended continuously to all convex compact sets in R^n, providing necessary conditions and explicit results in three dimensions.

Contribution

It identifies necessary conditions for the extendability of valuations by continuity and offers explicit results in R^3, advancing understanding of valuation extension.

Findings

Necessary condition for extension established

Explicit results obtained for R^3 case

Extension depends on initial valuation data

Abstract

There is a well known construction of weakly continuous valuations on convex compact polytopes in R^n. In this paper we investigate when a special case of this construction gives a valuation which extends by continuity in the Hausdorff metric to all convex compact subsets of R^n. It is shown that there is a necessary condition on the initial data for such an extension. In the case of R^3 more explicit results are obtained.