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Abstract

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.<br /><br />

We prove that to any comprehensive compact convex body &Gamma; , given a set of finitely many points on its surface, there is a Cephoid &Pi; that coincides with &Gamma; 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.<br /><br />

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|).<br /><br />

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|).<br /><br />

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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