Minkowski Endomorphisms

TL;DR

This paper advances the understanding of Minkowski endomorphisms by establishing their uniform continuity, exploring their monotonicity properties, improving valuation representation results, and clarifying structural limitations within specific classes.

Contribution

It proves all Minkowski endomorphisms are uniformly continuous, identifies non-monotone examples, enhances valuation representation under homogeneity, and shows the absence of McMullen decomposition in certain Minkowski valuations.

Findings

All Minkowski endomorphisms are uniformly continuous.

Existence of Minkowski endomorphisms that are not weakly-monotone.

No McMullen decomposition exists for certain continuous, even, $SO(n)$-equivariant Minkowski valuations.

Abstract

Several open problems concerning Minkowski endomorphisms and Minkowski valuations are solved. More precisely, it is proved that all Minkowski endomorphisms are uniformly continuous but that there exist Minkowski endomorphisms that are not weakly-monotone. This answers questions posed repeatedly by various authors. Furthermore, a recent representation result for Minkowski valuations by Schuster and Wannerer is improved under additional homogeneity assumptions. Also a question related to the structure of Minkowski endomorphisms by the same authors is answered. Finally, it is shown that there exists no McMullen decomposition in the class of continuous, even, $SO(n)$-equivariant and translation invariant Minkowski valuations extending a result by Parapatits and Wannerer.