# On the extendability by continuity of angular valuations on polytopes

## TL;DR

This paper investigates conditions under which translation-invariant, weakly continuous valuations on polytopes can be extended continuously to all convex bodies, providing a specific criterion in a particular case.

## Contribution

It offers a new criterion for extending certain valuations from polytopes to convex bodies, advancing understanding of valuation continuity and extendability.

## Key findings

- A simple necessary and sufficient condition for extension in a special case.
- Clarification of the relationship between weak continuity and full continuity.
- Insights into McMullen's valuation classification.

## Abstract

A classical theorem of P. McMullen describes all valuations on polytopes that are invariant under translations and weakly continuous, i.e., continuous with respect to parallel displacements of the facets of a polytope. While it is typically not difficult to check that a valuation is weakly continuous, it is not clear how to decide whether it admits a continuous extension to convex bodies. In a special case of McMullen's construction a simple necessary and sufficient condition on the initial data of such an extension is obtained.