Convex hyperspaces of probability measures and extensors in the asymptotic category

TL;DR

This paper constructs an example of an asymptotically zero-dimensional space where the space of compact convex probability measures fails to be an absolute extensor in the asymptotic category, challenging assumptions about extensibility.

Contribution

It provides the first example of a space with these properties, highlighting limitations in the extensibility of convex measure spaces within the asymptotic category.

Findings

The space of compact convex subsets of probability measures is not an absolute extensor.

An example of an asymptotically zero-dimensional space is constructed.

The result impacts understanding of extensors in the asymptotic category.

Abstract

The objects of the Dranishnikov asymptotic category are proper metric spaces and the morphisms are asymptotically Lipschitz maps. In this paper we provide an example of an asymptotically zero-dimensional space (in the sense of Gromov) whose space of compact convex subsets of probability measures is not an absolute extensor in the asymptotic category in the sense of Dranishnikov.