FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2007, VOLUME 13, NUMBER 4, PAGES 31-52

**Cyclic projectors and separation theorems in idempotent convex
geometry**

S. Gaubert

S. N. Sergeev

Abstract

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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.

Location: http://mech.math.msu.su/~fpm/eng/k07/k074/k07402h.htm

Last modified: November 28, 2007