On the polytopes of belief and plausibility functions

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In this paper we study some properties of the polytope of belief functions on a finite referential. These properties can be used in the problem of identification of a belief function from sample data. More concretely, we study the set of isometries, the set of invariant measures and the adjacency structure. From these results, we prove that the polytope of belief functions is not an order polytope if the referential has more than two elements. Similar results are obtained for plausibility functions.
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M. Grabisch, T. Murofushi and M. Sugeno (eds.), Fuzzy Measures and Integrals — Theory and Applications, No. 40 in Studies in Fuzziness and Soft Computing (Physica-Verlag, Heidelberg, Germany, 2000). M. Grabisch, k-order additive discrete fuzzy measures, in Proc. 6th Int. Conf. Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU), Granada, Spain, 1996, pp. 1345–1350. P. Miranda, M. Grabisch and P. Gil, p-symmetric fuzzy measures, Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 10(Suppl.) (2002) 105–123. E. F. Combarro and P. Miranda, On the polytope of non-additive measures, Fuzzy Sets and Systems 159(16) (2008) 2145–2162. P. Miranda and E. F. Combarro, On the structure of some families of fuzzy measures, IEEE Trans. Fuzzy Systems 15(6) (2007) 1068–1081. R. Stanley, Two poset polytopes, Discrete Comput. Geom. 1(1) (1986) 9–23. M. Sugeno, Theory of fuzzy integrals and its applications, PhD thesis, Tokyo Institute of Technology, 1974. G. C. Rota, On the foundations of combinatorial theory I. Theory of M¨obius functions, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 2 (1964)340–368. A. Chateauneuf and J.-Y. Jaffray, Some characterizations of lower probabilities and other monotone capacities through the use of M¨obius inversion, Mathematical Social Sciences 17 (1989) 263–283. A. N. Karkishchenko, Invariant fuzzy measures on a finite algebra, in Proc. North American Fuzzy Information Processing (NAPIF96), 1996, pp. 588–592. A. P. Dempster, Upper and lower probabilities induced by a multivalued mapping, The Annals of Mathematical Statististics 38 (1967) 325–339. G. Shafer, A Mathematical Theory of Evidence (Princeton University Press, Princeton, New Jersey, USA, 1976). P. Smets and R. Kennes, The transferable belief model, Artificial Intelligence 66 (1994) 191–234. F. Cuzzolin, A geometric approach to the Theory of Evidence, IEEE Trans. Systems, Man and Cybernetics-Part C: Applications and Reviews 38(4) (2008) 522–534. P. Miranda, E. F. Combarro and P. Gil, Extreme points of some families of nonadditive measures, European Journal of Operational Research 33(10) (2006) 3046–3066. G. Koshevoy, Distributive lattices and product of capacities, J. Mathematical Analysis and Applications 219 (1998) 427–441. E. F. Combarro and P. Miranda, Adjacency on the order polytope with applications to the theory of fuzzy measures, Fuzzy Sets and Systems 161(5) (2010) 619–641. E. F. Combarro and P. Miranda, Characterizing isometries on the order polytope with an application to the theory of fuzzy measures, Information Sciences 180 (2010)384–398. E. F. Combarro and P. Miranda, Identification of fuzzy measures from sample data with genetic algorithms, Computers and Operations Research 33(10) (2006) 3046–3066