Approximating Convex Bodies by Cephoids
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Joachim Rosenmüller: Center for Mathematical Economics, Bielefeld University
No 640, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University
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 Γ , given a set of finitely many points on its surface, there is a Cephoid Π that coincides with Γ 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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