Approximating Convex Bodies by Cephoids

Joachim Rosenmüller
Additional contact information
Joachim Rosenmüller: Center for Mathematical Economics, Bielefeld University

No 640, Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University

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. 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.

Pages: 28
Date: 2020-10-12
New Economics Papers: this item is included in nep-gth
References: Add references at CitEc
Citations:

Downloads: (external link)
https://pub.uni-bielefeld.de/download/2946529/2946530 First Version, 2020 (application/pdf)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:bie:wpaper:640

Access Statistics for this paper

More papers in Center for Mathematical Economics Working Papers from Center for Mathematical Economics, Bielefeld University Contact information at EDIRC.
Bibliographic data for series maintained by Bettina Weingarten ().