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

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 ⩽ ω 1 . It is also proved that the hyperspace of convex compact subsets of the Tychonov cube I ω 1 is homeomorphic to I ω 1 . An analogous result is also proved for the cone over I ω 1 . Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.