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Concentration of the Intrinsic Volumes of a Convex Body

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2266))

Abstract

The intrinsic volumes are measures of the content of a convex body. This paper applies probabilistic and information-theoretic methods to study the sequence of intrinsic volumes. The main result states that the intrinsic volume sequence concentrates sharply around a specific index, called the central intrinsic volume. Furthermore, among all convex bodies whose central intrinsic volume is fixed, an appropriately scaled cube has the intrinsic volume sequence with maximum entropy.

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Acknowledgements

We are grateful to Emmanuel Milman for directing us to the literature on concentration of information. Dennis Amelunxen, Sergey Bobkov, and Michel Ledoux also gave feedback at an early stage of this project. Ramon Van Handel provided valuable comments and citations, including the fact that ULC sequences concentrate. We thank the anonymous referee for a careful reading and constructive remarks.

Parts of this research were completed at Luxembourg University and at the Institute for Mathematics and its Applications (IMA) at the University of Minnesota. Giovanni Peccati is supported by the internal research project STARS (R-AGR-0502-10) at Luxembourg University. Joel A. Tropp gratefully acknowledges support from ONR award N00014-11-1002 and the Gordon and Betty Moore Foundation.

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, UK

    Martin Lotz

  2. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

    Michael B. McCoy

  3. Cruise LLC, San Francisco, CA, USA

    Michael B. McCoy

  4. Faculty of Science, Technology and Medicine, University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Ivan Nourdin & Giovanni Peccati

  5. Department of Mathematics, University of Luxembourg, Esch-sur-Alzette, Luxembourg

    Ivan Nourdin & Giovanni Peccati

  6. Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA, USA

    Joel A. Tropp

Authors
  1. Martin Lotz
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  2. Michael B. McCoy
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  3. Ivan Nourdin
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  4. Giovanni Peccati
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  5. Joel A. Tropp
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Corresponding author

Correspondence to Joel A. Tropp .

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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel; Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Bo'az Klartag

  2. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Lotz, M., McCoy, M.B., Nourdin, I., Peccati, G., Tropp, J.A. (2020). Concentration of the Intrinsic Volumes of a Convex Body. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2266. Springer, Cham. https://doi.org/10.1007/978-3-030-46762-3_6

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