Bi-Convexity and Bi-Martingales

Robert J. Aumann and Sergiu Hart

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A set in a product space X x Y is bi-convex if all its x- and y-sections are convex. A bi-martingale is a martingale with values in X x Y whose x- and y-coordinates change only one at a time. This paper investigates the limiting behavior of bimartingales in terms of the bi-convex hull of a set -- the smallest bi-convex set containing it -- and of several related concepts generalizing the concept of separation to the bi-convex case.