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 ⩽ω1. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω1 is homeomorphic to Iω1. An analogous result is also proved for the cone over Iω1. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.