A Riemannian interpolation inequality à la Borell, Brascamp and Lieb

By Dario Cordero-Erausquin, Robert J. McCann, M. Schmuckenschläger. 2001

375 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5fd94b2d3447557218df6eea51f13d83fccfca24

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Citations in the corpus

A survey of the Schr\"odinger problem and some of its connections with optimal transport (2013, background): “Sturm and M-K. von Renesse, Transport inequalities, gradient estimates, entropy, and Ricci curvature, Comm. Pure Appl.”
Five lectures on optimal transportation: Geometry, regularity and applications (2010, result): “One direction of this equivalence was proved by Cordero-Erausquin, McCann, and Schmuckenschäger [30] and Otto and Villani [110] in projects which were initially independent (see also [28] and [31]), while the converse was established by von Renesse and Sturm [116] confirming the formal arguments of ”
On the role of convexity in functional and isoperimetric inequalities (2008, background): “rom which the same conclusion will hold for ψ x0, and consequently (6.1) will hold for ψ. In what follows, we refer to Greene and Wu [28], Bangert [3], CorderoErausquin, McCann and Schmuckenschla¨ger [24], and the references therein, for further information on geodesically convex and semi-convex fun”
Nonlinear Diffusion Equations and Curvature Conditions in Metric Measure Spaces (2015, background): “In the framework of optimal transport, the importance of curvature bounds has been deeply analyzed in [44, 26, 53].”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [98] [73] [74] [30] [51], geometry [83] [29] [95] [71] [70] [104] [79] [63], dynamics [7] [10] [59] [82] and nonlinear partial differential equations [18] [20] [21] [36] [42] [72] [102], as well”
Stochastic Gradient Descent on Riemannian Manifolds (2011, result): “If the sectional curvature is lower bounded by κ < 0 we have, according to a recent result [15]”
Ricci curvature for metric-measure spaces via optimal transport (2004, methodology): “In the past few years, some regularity results for optimal transport on R have been extended to Riemannian manifolds [18, 29].”
On the role of convexity in isoperimetry, spectral gap and concentration (2007, methodology): “ith-density setting, using the following remarkable generalization of the Prekopa-Leindler inequality due to Cordero-Erausquin, McCann and Schmuckenschlager [35] (generalizing their own result from [34]). Given x;y2Mand s2[0;1], dene: Z s(x;y) := fz2M;d(x;z) = sd(x;y) and d(z;y) = (1 s)d(x;y)g: Theo”
Finsler interpolation inequalities (2009, methodology): “See [ 11 ,15] for their correlation and historical background.”
Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds (2012, background): “The importance of curvature bounds in the framework of optimal transport has been deeply analyzed in [45, 21, 56].”
On the regularity of solutions of optimal transportation problems (2009, background): “of optimal maps in that case was established by McCann [25], and further examined by several authors, with many interesting applications in geometric and functional analysis (for example [ 12 ] and [26]).”
Characterization of absolutely continuous curves in Wasserstein spaces (2006, background): “Indeed, many papers have recently appeared dealing with various aspects of measure metric spaces or, in particular, of Riemannian manifolds, strictly connected to Kantorovitch–Rubinstein–Wasserstein distance (see [21,22,30,31], in metric measure spaces, and [14,25,26,29,33]).”