Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

TL;DR

This paper characterizes when the hyperspace of convex compact subsets of a compact convex set in a locally convex space is an absolute retract, linking it to the space's weight, and explores its topological properties.

Contribution

It establishes a characterization of hyperspaces as absolute retracts based on the weight of the original space and proves homeomorphism results for hyperspaces of the Tychonov cube.

Findings

Hyperspace of convex compact subsets is an absolute retract iff the original space has weight ≤ ω₁.

Hyperspace of convex compact subsets of the Tychonov cube I^{ω₁} is homeomorphic to I^{ω₁}.

Selection theorems for maps of hyperspaces are developed.

Abstract

Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\le\omega_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{\omega_1}$ is homeomorphic to $I^{\omega_1}$. An analogous result is also proved for the cone over $I^{\omega_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.