  Let X be a Tychonoff space and MC(X) be the space of convex minimal usco maps
with values in R, the space of real numbers. Such set-valued maps are important
in the study of subdifferentials of convex functions. Using the strong Choquet
game we prove complete metrizability of MC(X) with the upper Vietoris topology.
If X is normal, elements of MC(X) can be approximated in the Vietoris topology
by continuous functions. We also study first countability, second countability
and other properties of the upper Vietoris topology on MC(X).
