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Geometric Aspects of Functional Analysis pp 49–61Cite as

Bobkov’s Inequality via Optimal Control Theory

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2256))

Abstract

We give a Bellman proof of Bobkov’s inequality using arguments of dynamic programming. As a byproduct of the method we obtain a characterization of smooth optimizers.

This paper is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while two of the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester. P.I. is partially supported by NSF DMS-1856486 and CAREER DMS-1945102.

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Notes

  1. 1.

    Here B appears as an infimum instead of supremum, but then − B is a supremum and the above reasoning applies in the same way, except that all inequalities will be reversed and \(\sup \) in (3.2.6) will be replaced by \(\inf \).

  2. 2.

    Notice that \(\varphi ^{2}(t)-B_{x}^{2} \neq 0\) follows from the second equation in (3.3.9)

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Acknowledgement

We would like to thank the anonymous referee for his/her helpful suggestions and comments.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, Toulouse Cedex 9, France

    Franck Barthe

  2. University of California Irvine, Irvine, CA, USA

    Paata Ivanisvili

Authors
  1. Franck Barthe
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  2. Paata Ivanisvili
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Correspondence to Franck Barthe .

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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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Barthe, F., Ivanisvili, P. (2020). Bobkov’s Inequality via Optimal Control Theory. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2256. Springer, Cham. https://doi.org/10.1007/978-3-030-36020-7_3

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