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

Carathéodory’s, Helly’s and Radon’s theorems are three basic results in discrete geometry. Their max-plus counterparts have been proved by various authors. In this paper, more advanced results in discrete geometry are shown to have also their max-plus counterparts: 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 settings.