On optimal transport maps between $\frac{1}{d}$-concave densities

In this paper we extend the scope of Caffarelli’s contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in \mathbb{R}^{d} . Our focus is on a broader category of densities, specifically those that are \frac{1}{d} -concave and can be represented as V^{-d} , where V is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli’s theorem.

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Publication Details

Journal
Annales de l Institut Henri Poincaré C Analyse Non Linéaire
Published
2024-11-07
DOI
https://doi.org/10.4171/aihpc/148
Citations
1
Primary Topic
Caveolin-1 and cellular processes
Type
article
Field-Weighted Citation Impact
0.33
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article

On optimal transport maps between $\frac{1}{d}$-concave densities

Alessio Figalli, Filippo Santambrogio, Guillaume Carlier
1 citations
Annales de l Institut Henri Poincaré C Analyse Non Linéaire
Caveolin-1 and cellular processes
0.33
article

On optimal transport maps between $\frac{1}{d}$-concave densities

Alessio Figalli, Filippo Santambrogio, Guillaume Carlier
article en
1 citations

Abstract

In this paper we extend the scope of Caffarelli’s contraction theorem, which provides a measure of the Lipschitz constant for optimal transport maps between log-concave probability densities in \mathbb{R}^{d} . Our focus is on a broader category of densities, specifically those that are \frac{1}{d} -concave and can be represented as V^{-d} , where V is convex. By setting appropriate conditions, we derive linear or sublinear limitations for the optimal transport map. This leads us to a comprehensive Lipschitz estimate that aligns with the principles established in Caffarelli’s theorem.

Annales de l Institut Henri Poincaré C Analyse Non LinéaireVol. 43(2)
Université Claude Bernard Lyon 1 (FR), École Centrale de Lyon (FR), Centre National de la Recherche Scientifique (FR), Institut national de recherche en sciences et technologies du numérique (FR), ETH Zurich (CH), Institut Camille Jordan (FR), Institut National des Sciences Appliquées de Lyon (FR), Université Paris Dauphine-PSL (FR), Université Jean Monnet (FR)
Openalex Percentile: Top 39%
Caveolin-1 and cellular processes
0.33
DOIOpen Access PDFOpenAlex
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On optimal transport maps between $\frac{1}{d}$-concave densities — Alessio Figalli, Filippo Santambrogio, et al. · Annales de l Institut Henri Poincaré C Analyse Non Linéaire (2024) | TGRS Research Map | TGRS