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The Borell–Ehrhard game

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. 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.

Author information

Correspondence to Ramon van Handel.

Additional information

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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Keywords

  • Gaussian measures
  • Convexity
  • Ehrhard inequality
  • Stochastic games

Mathematics Subject Classification

  • 60G15
  • 39B62
  • 52A40
  • 91A15