The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg

Jesús A. de Loera (1) , Xavier Goaoc (2) , Frédéric Meunier (3) , Nabil Mustafa (4)

1 UC Davis - University of California [Davis]
2 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
3 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
4 LIGM - Laboratoire d'Informatique Gaspard-Monge

We discuss five discrete results: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carath\'eodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as game theory, graph theory, mathematical optimization, computational geometry, etc.

Domaines: Mathématiques [math]

Jesús A. de Loera, Xavier Goaoc, Frédéric Meunier, Nabil Mustafa. The discrete yet ubiquitous theorems of Caratheodory, Helly, Sperner, Tucker, and Tverberg. Bulletin of the American Mathematical Society, 2019, 56, pp.415-511. ⟨10.1090/bull/1653⟩. ⟨hal-02050466⟩