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This article is cited in 16 scientific papers (total in 16 papers)
Cyclic projectors and separation theorems in idempotent convex geometry
S. Gauberta, S. N. Sergeevb
a French National Institute for Research in Computer Science and Automatic Control,
INRIA Paris - Rocquencourt Research Centre
b M. V. Lomonosov Moscow State University
Abstract:
Semimodules over idempotent semirings like the max-plus or tropical semiring have much in common with convex cones. This analogy is particularly apparent in the case of subsemimodules of the $n$-fold Cartesian product of the max-plus semiring: It is known that one can separate
a vector from a closed subsemimodule that does not contain it. Here we establish a more general separation theorem, which applies to any finite collection of closed subsemimodules with a trivial intersection. The proof of this theorem involves specific nonlinear operators, called here
cyclic projectors on idempotent semimodules. These are analogues of the cyclic nearest-point projections known in convex analysis. We obtain a theorem that characterizes the spectrum of cyclic projectors on idempotent semimodules in terms of a suitable extension of Hilbert's projective metric. We also deduce as a corollary of our main results the idempotent analogue of Helly's theorem.
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Journal of Mathematical Sciences (New York), 2008, 155:6, 815–829
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UDC:
512.643+512.558
Citation:
S. Gaubert, S. N. Sergeev, “Cyclic projectors and separation theorems in idempotent convex geometry”, Fundam. Prikl. Mat., 13:4 (2007), 31–52; J. Math. Sci., 155:6 (2008), 815–829
Citation in format AMSBIB
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\jour Fundam. Prikl. Mat.
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\pages 31--52
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\jour J. Math. Sci.
\yr 2008
\vol 155
\issue 6
\pages 815--829
\crossref{https://doi.org/10.1007/s10958-008-9243-8}
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This publication is cited in the following articles:
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Gaubert S., Katz R.D., “The tropical analogue of polar cones”, Linear Algebra Appl., 431:5-7 (2009), 608–625
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Akian M., Gaubert S., Guterman A., “Linear independence over tropical semirings and beyond”, Tropical and idempotent mathematics, Contemp. Math., 495, Amer. Math. Soc., Providence, RI, 2009, 1–38
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Sergeev S., “Multiorder, Kleene stars and cyclic projectors in the geometry of max cones”, Tropical and Idempotent Mathematics, Contemp. Math., 495, Amer. Math. Soc., Providence, RI, 2009, 317–342
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Gaubert S., “Max-plus algebraic tools for discrete event systems, static analysis, and zero-sum games”, Formal modeling and analysis of timed systems, Lecture Notes in Computer Science, 5813, Springer, Berlin, 2009, 7–11
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Gaubert S., Meunier F., “Carathéodory, Helly and the others in the max-plus world”, Discrete Comput. Geom., 43:3 (2010), 648–662
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Singer I., “Elementary topical functions on $b$-complete semimodules over $b$-complete idempotent semifields”, Linear Algebra Appl., 433:11–12 (2010), 2139–2146
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Nitica V., Sergeev S., “On hyperplanes and semispaces in max-min convex geometry”, Kybernetika (Prague), 46:3 (2010), 548–557
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Gaubert S., Katz R.D., “Minimal half-spaces and external representation of tropical polyhedra”, J. Algebr. Comb., 33:3 (2011), 325–348
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Allamigeon X., Gaubert S., Katz R.D., “The number of extreme points of tropical polyhedra”, J. Combin. Theory Ser. A, 118:1 (2011), 162–189
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Akian M., Gaubert S., Niţică V., Singer I., “Best approximation in max-plus semimodules”, Linear Algebra Appl., 435:12 (2011), 3261–3296
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Nitica V., Sergeev S., “An interval version of separation by semispaces in max-min convexity”, Linear Algebra Appl., 435:7 (2011), 1637–1648
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Akian M., Gaubert S., Guterman A., “Tropical polyhedra are equivalent to mean payoff games”, Internat. J. Algebra Comput., 22:1 (2012), 1250001, 43 pp.
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Singer I., Nitica V., “Topical Functions on Semimodules and Generalizations”, Linear Alg. Appl., 437:10 (2012), 2471–2488
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Allamigeon X., Gaubert S., Goubault E., “Computing the Vertices of Tropical Polyhedra Using Directed Hypergraphs”, Discret. Comput. Geom., 49:2 (2013), 247–279
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Durcheva M.I., “An Application of Different Dioids in Public Key Cryptography”, Applications of Mathematics in Engineering and Economics (AMEE'14), AIP Conference Proceedings, 1631, eds. Venkov G., Pasheva V., Amer Inst Physics, 2014, 336–343
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Hassani S., Mohebi H., “Characterizations of Minimal Elements of Topical Functions on Semimodules With Applications”, Linear Alg. Appl., 520 (2017), 104–124
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