Vol. 83, No. 2, 1979

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Hyperspaces of compact convex sets

Sam Bernard Nadler, Jr., Joseph E. Quinn and N. Stavrakas

Vol. 83 (1979), No. 2, 441–462

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.

Mathematical Subject Classification 2000
Primary: 57N20
Secondary: 46A55
Received: 2 August 1976
Revised: 4 February 1977
Published: 1 August 1979
Sam Bernard Nadler, Jr.
Joseph E. Quinn
N. Stavrakas