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Geometric Aspects of Functional Analysis pp 231–260Cite as

Logarithmically-Concave Moment Measures I

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2116))

Abstract

We discuss a certain Riemannian metric, related to the toric Kähler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in \(\mathbb{R}^{n}\). We use this metric in order to bound the second derivatives of the solution to the toric Kähler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.

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Acknowledgements

The author would like to thank Bo Berndtsson, Dario Cordero-Erausquin, Ronen Eldan, Alexander Kolesnikov, Eveline Legendre, Emanuel Milman, Ron Peled, Yanir Rubinstein and Boris Tsirelson for interesting discussions related to this work. Supported by a grant from the European Research Council (ERC).

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Authors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

    Bo’az Klartag

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  1. Bo’az Klartag
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Correspondence to Bo’az Klartag .

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Editors and Affiliations

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

    Bo'az Klartag

  2. Technion - Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Klartag, B. (2014). Logarithmically-Concave Moment Measures I. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_16

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