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Analysis at Large pp 203–231Cite as

The Slicing Problem by Bourgain

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Abstract

In the context of his work on maximal functions in the 1980s, Jean Bourgain came across the following geometric question: Is there c > 0 such that for any dimension n and any convex body \(K \subseteq \mathbb R^n\) of volume one, there exists a hyperplane H such that the (n − 1)-dimensional volume of K ∩ H is at least c? This innocent and seemingly obvious question (which remains unanswered!) has established a new direction in high-dimensional geometry. It has emerged as an “engine” that inspired the discovery of many deep results and unexpected connections. Here we provide a survey of these developments, including many of Bourgain’s results.

Dedicated to the memory of Jean Bourgain

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Authors and Affiliations

  1. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    B. Klartag

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    V. Milman

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  1. B. Klartag
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  2. V. Milman
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Editors and Affiliations

  1. Institute of Mathematics, University of Zurich, Zürich, Switzerland

    Artur Avila

  2. Department of Mathematics and Engineering Sciences, Hellenic Military Academy, Athens, Greece

    Michael Th. Rassias

  3. Department of Mathematics, Princeton University, Princeton, NJ, USA

    Yakov Sinai

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Klartag, B., Milman, V. (2022). The Slicing Problem by Bourgain. In: Avila, A., Rassias, M.T., Sinai, Y. (eds) Analysis at Large. Springer, Cham. https://doi.org/10.1007/978-3-031-05331-3_9

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