Mathematical Proceedings of the Cambridge Philosophical Society

Research Article

Measures of the non-convexity of sets and the Shapley–Folkman–Starr theorem

J. W. S. Casselsa1

a1 Department of Pure Mathematics and Mathematical Statistics, Cambridge

The object of this note is to show that elementary probability considerations suggest a very natural way of measuring the non-convexity of a set in euclidean space or, more generally, in a real Hilbert space S0305004100051884_xs1D573. In particular they give a proof, much simpler and under less restrictive conditions, of results due to Shapley, Folkman and Starr which are of importance in Mathematical Economics ((1),(2)).

(Received April 03 1975)