The purpose of this paper is to
develop in detail certain aspects of the space of nonempty compact convex subsets of
a subset X (denoted cc(X)) of a metric locally convex T.V.S. It is shown that if X is
compact and dim(X) ≧ 2 then cc(X) is homeomorphic with the Hilbert cube
(denoted cc(X)≅I∞). It is shown that if n ≧ 2, then cc(Rn) is homeomorphic to
I∞ with a point removed. More specialized results are that if X ⊂ R2 is
such that cc(X)≅I∞ then X is a two cell; and that if X ⊂ R3 is such that
cc(X)≅I∞ and X is not contained in a hyperplane then X must contain a three
cell.
For the most part we will be restricting ourselves to compact spaces X although
in the last section of the paper, §7, we consider some fundamental noncompact
spaces.
