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

  • J. W. S. Cassels (a1)

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 . 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)).

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(1)Arrow, K. J. and Hahn, F. H.General competitive analysia. (Holden Day, San Francisco and Oliver and Boyd, Edinburgh 1971, especially chapters 7, 8 and Appendix B.)
(2)Starr, R.Quasi-equilibria in markets with nonconvex preferences. Econometrics 37 (1969), 2538.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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