Carathéodory, Helly and the Others in the Max-Plus World

Authors:
Stéphane Gaubert;Frédéric Meunier
Affiliations:
École Polytechnique, INRIA and CMAP, 91128, Palaiseau cedex, France;ENPC, Université Paris Est, LVMT, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455, Marne-la-Vallée cedex 2, France
Venue:
Discrete & Computational Geometry
Year:
2010

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Abstract

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus or tropical analogues have been proved by various authors. We show that more advanced results in discrete geometry also have max-plus analogues, namely, the colorful Carathéodory theorem and the Tverberg theorem. A conjecture connected to the Tverberg theorem—Sierksma’s conjecture—although still open for the usual convexity, is shown to be true in the max-plus setting.