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


Authors: Jesús A. De Loera, Xavier Goaoc, Frédéric Meunier and Nabil H. Mustafa
Journal: Bull. Amer. Math. Soc. 56 (2019), 415-511
MSC (2010): Primary 52Cxx, 57M99, 90Cxx, 91Axx
DOI: https://doi.org/10.1090/bull/1653
Published electronically: January 25, 2019
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Abstract: We discuss five fundamental results of discrete mathematics: the lemmas of Sperner and Tucker from combinatorial topology and the theorems of Carathéodory, Helly, and Tverberg from combinatorial geometry. We explore their connections and emphasize their broad impact in application areas such as data science, game theory, graph theory, mathematical optimization, computational geometry, etc.


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Additional Information

Jesús A. De Loera
Affiliation: University of California, Department of Mathematics, Davis, California 95616
Email: deloera@math.ucdavis.edu

Xavier Goaoc
Affiliation: Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
Email: xavier.goaoc@loria.fr

Frédéric Meunier
Affiliation: Université Paris Est, CERMICS, ENPC, F-77454, Marne-la-Vallée, France
Email: frederic.meunier@enpc.fr

Nabil H. Mustafa
Affiliation: Université Paris-Est, LIGM, Equipe A3SI, ESIEE Paris, Noisy le-Grand, France
Email: nabilhassan.mustafa@esiee.fr

DOI: https://doi.org/10.1090/bull/1653
Received by editor(s): June 16, 2018
Published electronically: January 25, 2019
Additional Notes: The first author was partially supported by LabEx Bezout grant ANR-10-LABX-58 and also by NSF grant DMS-1522158.
The second author was partially supported by Institut Universitaire de France.
The fourth author was supported by ANR SAGA grant JCJC-14-CE25-0016-01.
Article copyright: © Copyright 2019 American Mathematical Society