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no. 5
Spectral monotonicity under Gaussian convolution
Bo’az Klartag1; Eli Putterman2
1 Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
2 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 939-967.

We show that the Poincaré constant of a log-concave measure in Euclidean space is monotone increasing along the heat flow. In fact, the entire spectrum of the associated Laplace operator is monotone decreasing. Two proofs of these results are given. The first proof analyzes a curvature term of a certain time-dependent diffusion, and the second proof constructs a contracting transport map following the approach of Kim and Milman.

Nous montrons que la constante de Poincaré d’une mesure log-concave dans l’espace euclidien est croissante le long du flot de la chaleur. En fait, le spectre entier de l’opérateur de Laplace associé est décroissant. Deux preuves de ces résultats sont données. La première preuve analyse un terme de courbure d’une certaine diffusion dépendant du temps, et la seconde preuve construit une application de transport contractante en suivant l’approche de Kim et Milman.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1759

Bo’az Klartag 1; Eli Putterman 2

1 Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel
2 School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Bo’az Klartag; Eli Putterman. Spectral monotonicity under Gaussian convolution. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 32 (2023) no. 5, pp. 939-967. doi : 10.5802/afst.1759. https://afst.centre-mersenne.org/articles/10.5802/afst.1759/

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