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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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The discrete yet ubiquitous theorems of Carathéodory, Helly, Sperner, Tucker, and Tverberg
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by Jesús A. De Loera, Xavier Goaoc, Frédéric Meunier and Nabil H. Mustafa PDF
Bull. Amer. Math. Soc. 56 (2019), 415-511 Request permission


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
  • MR Author ID: 364032
  • ORCID: 0000-0002-9556-1112
  • Email:
  • Xavier Goaoc
  • Affiliation: Université de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France
  • MR Author ID: 729132
  • Email:
  • Frédéric Meunier
  • Affiliation: Université Paris Est, CERMICS, ENPC, F-77454, Marne-la-Vallée, France
  • MR Author ID: 698556
  • Email:
  • Nabil H. Mustafa
  • Affiliation: Université Paris-Est, LIGM, Equipe A3SI, ESIEE Paris, Noisy le-Grand, France
  • MR Author ID: 737827
  • ORCID: 0000-0003-1046-6157
  • Email:
  • 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.
  • © Copyright 2019 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 56 (2019), 415-511
  • MSC (2010): Primary 52Cxx, 57M99, 90Cxx, 91Axx
  • DOI:
  • MathSciNet review: 3974609