This article is cited in 23 papers

Cyclic projectors and separation theorems in idempotent convex geometry

S. Gaubert^a, S. N. Sergeev<sup>b

a</sup> 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.

UDC: 512.643+512.558

 English version: Journal of Mathematical Sciences (New York), 2008, 155:6, 815–829

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