Logarithmically-concave moment measures I

Boaz Klartag

doi:10.1007/978-3-319-09477-9_16

Logarithmically-concave moment measures I

Research output: Chapter in Book/Report/Conference proceeding › Chapter

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 Rⁿ. 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.

Original language	English
Title of host publication	Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics
Editors	Boaz Klartag, Emanuel Milman
Place of Publication	Switzerland
Pages	231-260
Number of pages	30
Volume	2116
ISBN (Electronic)	978-3-319-09477-9
DOIs	https://doi.org/10.1007/978-3-319-09477-9_16
State	Published - 2014
Externally published	Yes

Publication series

Name	Lecture Notes in Mathematics
ISSN (Print)	0075-8434

All Science Journal Classification (ASJC) codes

Algebra and Number Theory

Access to Document

10.1007/978-3-319-09477-9_16

http://arxiv.org/abs/1309.2767License: Other

Cite this

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title = "Logarithmically-concave moment measures I",

abstract = "We discuss a certain Riemannian metric, related to the toric K{\"a}hler-Einstein equation, that is associated in a linearly-invariant manner with a given log-concave measure in Rn. We use this metric in order to bound the second derivatives of the solution to the toric K{\"a}hler-Einstein equation, and in order to obtain spectral-gap estimates similar to those of Payne and Weinberger.",

author = "Boaz Klartag",

note = "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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N2 - 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 Rn. 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.

AB - 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 Rn. 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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