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On the polytopes of belief and plausibility functions. (English) Zbl 1211.28017

Belief functions (\(\infty\)-monotone capacities) on a finite set \(X\) form a polytope. In this paper, the authors study several interesting properties of this polytope, whose \(2^n-1\) vertices are just \(\{0,1\}\)-valued belief functions (\(n\) being the cardinality of \(X\)). The study of isometries (wrt. Euclidean distance of set functions), invariant measures and the adjacency structure has resulted into the main result of this paper, proving that the polytope of belief functions on \(X\) is not an order polytope whenever \(n> 2\). By duality, similar results for the plausibility functions are obtained.

MSC:

28E10 Fuzzy measure theory
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
68T37 Reasoning under uncertainty in the context of artificial intelligence
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References:

[1] Grabisch M., Studies in Fuzziness and Soft Computing, in: Fuzzy Measures and Integrals – Theory and Applications (2000)
[2] DOI: 10.1142/S0218488502001867 · Zbl 1068.28013 · doi:10.1142/S0218488502001867
[3] DOI: 10.1016/j.fss.2007.12.021 · Zbl 1177.28041 · doi:10.1016/j.fss.2007.12.021
[4] DOI: 10.1109/TFUZZ.2007.895953 · Zbl 05516343 · doi:10.1109/TFUZZ.2007.895953
[5] DOI: 10.1007/BF02187680 · Zbl 0595.52008 · doi:10.1007/BF02187680
[6] DOI: 10.1016/0165-4896(89)90056-5 · Zbl 0669.90003 · doi:10.1016/0165-4896(89)90056-5
[7] DOI: 10.1214/aoms/1177698950 · Zbl 0168.17501 · doi:10.1214/aoms/1177698950
[8] Shafer G., A Mathematical Theory of Evidence (1976) · Zbl 0326.62009
[9] DOI: 10.1016/0004-3702(94)90026-4 · Zbl 0807.68087 · doi:10.1016/0004-3702(94)90026-4
[10] DOI: 10.1109/TSMCC.2008.919174 · doi:10.1109/TSMCC.2008.919174
[11] Miranda P., European Journal of Operational Research 33 pp 3046–
[12] DOI: 10.1006/jmaa.1997.5830 · Zbl 0910.06007 · doi:10.1006/jmaa.1997.5830
[13] DOI: 10.1016/j.fss.2009.05.004 · Zbl 1188.28017 · doi:10.1016/j.fss.2009.05.004
[14] DOI: 10.1016/j.ins.2009.09.020 · Zbl 1227.05035 · doi:10.1016/j.ins.2009.09.020
[15] DOI: 10.1016/j.cor.2005.02.034 · Zbl 1086.90069 · doi:10.1016/j.cor.2005.02.034
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