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The Borell–Ehrhard game

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Abstract

A precise description of the convexity of Gaussian measures is provided by sharp Brunn–Minkowski type inequalities due to Ehrhard and Borell. We show that these are manifestations of a game-theoretic mechanism: a minimax variational principle for Brownian motion. As an application, we obtain a Gaussian improvement of Barthe’s reverse Brascamp–Lieb inequality.

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Notes

  1. After this paper was completed, the author learned of recent work [34] where another rather delicate proof of Ehrhard’s inequality is provided using the Ornstein-Uhlenbeck semigroup.

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Acknowledgements

The author is grateful to the anonymous referees for comments that helped improve the presentation of the paper.

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Authors and Affiliations

  1. Princeton University, Fine Hall 208, Princeton, NJ, 08544, USA

    Ramon van Handel

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  1. Ramon van Handel
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Correspondence to Ramon van Handel.

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Supported in part by NSF Grant CAREER-DMS-1148711 and by the ARO through PECASE Award W911NF-14-1-0094.

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van Handel, R. The Borell–Ehrhard game. Probab. Theory Relat. Fields 170, 555–585 (2018). https://doi.org/10.1007/s00440-017-0762-4

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