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Journal of Convex Analysis 22 (2015), No. 2, 537--540
Copyright Heldermann Verlag 2015

A Minmax Theorem for Concave-convex Mappings with no Regularity Assumptions

Vianney Perchet
Université Paris-Diderot, LPMA -- 8 place FM/13, 75013 Paris, France

Guillaume Vigeral
Université Paris-Dauphine, CEREMADE, Place Mar. Lattre de Tassigny, 75775 Paris Cedex 16, France

We prove that zero-sum games with a concave-convex payoff mapping defined on a product of convex sets have a value as soon as the payoff mapping is bounded and one of the set is bounded and finite dimensional. In particular, no additional regularity assumption is required, such as lower or upper semicontinuity of the function or compactness of the sets. We provide several examples that show that our assumptions are minimal.

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