On tameness of zonoids

TL;DR

This paper proves that within a certain class of convex bodies, the subset of zonoids can be described using a logical framework called o-minimal structures, highlighting their definability and structural properties.

Contribution

It establishes that the set of zonoids in a globally subanalytic family is log-analytic and definable in an o-minimal structure, linking convex geometry with model theory.

Findings

Zonoids form a log-analytic set within globally subanalytic families.

The set of zonoids is definable in an o-minimal structure generated by exponential and globally subanalytic sets.

The result connects convex geometry with logical definability and tame topology.

Abstract

We prove that in a globally subanalytic family of convex bodies the set of zonoids is log-analytic, and in particular it is definable in the o-minimal structure generated by globally subanalytic sets and the graph of the exponential function.