Cyclic projectors and separation theorems in idempotent convex geometry

TL;DR

This paper develops a general separation theorem for closed semimodules over idempotent semirings, using spectral analysis of idempotent cyclic projectors, extending classical convex geometry results to the tropical setting.

Contribution

It introduces a new separation theorem for semimodules in idempotent convex geometry and analyzes the spectral properties of idempotent cyclic projectors.

Findings

Established a general separation theorem for closed semimodules with trivial intersection.

Characterized the spectrum of idempotent cyclic projectors using an extension of Hilbert's projective metric.

Derived an idempotent analogue of Helly's theorem.

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.