Approximating convex bodies by Cephoids / Joachim Rosenmüller
VerfasserRosenmüller, Joachim
ErschienenBielefeld, Germany : Center for Mathematical Economics (IMW), Bielefeld University, Oktober 2020
Elektronische Ressource
Umfang1 Online-Ressource (28 Seiten) : Illustrationen
SerieCenter for Mathematical Economics Working papers ; 640
SchlagwörterMinkowski-Summe / Konvexer Körper
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Approximating convex bodies by Cephoids [0.41 mb]

We consider a class of comprehensive compact convex polyhedra called Cephoids. A Cephoid is a Minkowski sum of finitely many standardized simplices ("deGua Simplices"). The Pareto surface of Cephoids consists of certain translates of simplices, algebraic sums of subsimplices etc. The peculiar shape of such a Pareto surface raises the question as to how far results for Cephoids can be carried over to general comprehensive compact convex bodies by approximation. We prove that to any comprehensive compact convex body , given a set of finitely many points on its surface, there is a Cephoid that coincides with in exactly these preset points. As a consequence, Cephoids are dense within the set of comprehensive compact convex bodies with respect to the Hausdorff metric. Cephoids appear in Operations Research (Optimization |10|, |3|), in Mathematical Economics (Free Trade theory |7|, |8|), and in Cooperative Game Theory (the Maschler--Perles solution |6|). More generally in the context of Cooperative Game Theory, the notion of a Cephoid serves to construct "solutions'' or "values'' for bargaining problems and non--side payment games (|9|). Therefore, the results of this paper open up an avenue for the extension of solution concepts from Cephoids to general compact convex bodies.

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