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Journal of Convex Analysis 11 (2004), No. 1, 209--234
Copyright Heldermann Verlag 2004

On the Regularity of the Convexification Operator on a Compact Set

Rida Laraki
CNRS, Lab. d'Econometrie de l'Ecole Polytechnique, 1 rue Descartes, 75005 Paris, France,

Let coX( . ) denote the convexification operator on bounded real functions on a convex compact set X. Several necessary and sufficient conditions for the operator coX( . ) to preserve continuity and uniformly Lipschitz continuity are established.
In the special case of a finite dimensional topological vector space, it is shown that (1) the preservation of continuity is equivalent to the closeness of the set of faces of X and (2) the uniform preservation of Lipschitz continuity is equivalent to X being a polytope.

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