On the inverse Klain map

TL;DR

This paper studies the inverse Klain map's continuity and characterizes convex bodies where valuations depend continuously on their Klain functions, revealing limitations in McMullen's decomposition for certain valuations.

Contribution

It characterizes convex bodies with continuous valuation dependence on Klain functions and proves the non-existence of McMullen's decomposition for specific Minkowski valuations.

Findings

Identifies classes of convex bodies with continuous valuation dependence

Shows McMullen's decomposition does not hold for certain valuations

Solves a longstanding problem posed by Schneider and Schuster

Abstract

The continuity of the inverse Klain map is investigated and the class of centrally symmetric convex bodies at which every valuation depends continuously on its Klain function is characterized. Among several applications, it is shown that McMullen's decomposition is not possible in the class of translation-invariant, continuous, positive valuations. This implies that there exists no McMullen decomposition for translation-invariant, continuous Minkowski valuations, which solves a problem first posed by Schneider and Schuster.