Spaces of idempotent measures of compact metric spaces

We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube

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 $\le\omega_1$. 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

Spaces of idempotent measures of compact metric spaces, Topol

a r t i c l e i n f o a b s t r a c t We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube