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arXiv · 0706.3347

Cyclic projectors and separation theorems in idempotent convex geometry

Stephane Gaubert·Sergei Sergeev
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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. We establish here a more general separation theorem, which applies to any finite collection of closed semimodules with a trivial intersection. In order to prove this theorem, we investigate the spectral properties of certain nonlinear operators called here idempotent cyclic projectors. These are idempotent analogues of the cyclic nearest-point projections known in convex analysis. The spectrum of idempotent cyclic projectors is characterized in terms of a suitable extension of Hilbert's projective metric. We deduce as a corollary of our main results the idempotent analogue of Helly's theorem.

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Stephane Gaubert, Sergei Sergeev. 2007-06-22. Cyclic projectors and separation theorems in idempotent convex geometry. https://doi.org/10.1007/s10958-008-9243-8

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