A Convexity Principle for Interacting Gases

By Robert J. McCann. 1997

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Flow Matching for Generative Modeling (2022, positive): “Furthermore, this family of probability paths also includes a particularly interesting case: the vector field that corresponds to an Optimal Transport (OT) displacement interpolant (McCann, 1997).”
Computational Optimal Transport (2018, positive): “An important result of McCann [1997] is that generalized “entropy” functions of the form F (α) = ∫ Rd φ(ρα(x))dx on X = Rd are geodesically convex if φ is convex, with φ(0) = 0, φ(t)/t→ +∞ as t→ +∞ and such that s 7→ sdφ(s−d) is convex decaying.”
Alexandrov meets Lott-Villani-Sturm (2010, background): “convexityofcertain functionals in the Wassersteinspace over a Riemannian manifold, appears in [Otto–Villani], [CMS], [Sturm–v.Renesse]. In the Euclidean context, this notion of convexity goes back to [McCann]. More on the history of the subject can be found in [Villani]. For a metric-measure space X”
Sharp Gagliardo–Nirenberg Inequalities via p-Laplacian Type Equations (2008, unknown): “Here F : [0,∞) → R satisfies F (0) = 0 and x → xF (x−n) is convex and nonincreasing (see McCann [11]), PF (x) = xF ′(x) − F (x), H |x|p∗/p∗(ρ) = H F (ρ) + H|x|p∗/p∗(ρ) = ∫ Rn F (ρ(x)) dx + ∫ Rn |x|p p∗ ρ(x)dx, and P(Rn) denotes the set of probability densities on R with finite p∗-moment.”
Characterization of absolutely continuous curves in Wasserstein spaces (2006, background): “[23] where the convex functionals in P2(R) was firstly studied and [21,29]).”
Balls have the worst best Sobolev inequalities (2005, background): “By Brenier’s Theorem [3], or rather the slightly more general version established by McCann (see [12] or [14, Corollary 2.”
Sharp weighted Sobolev and Gagliardo–Nirenberg inequalities on half‐spaces via mass transport and consequences (2013, background): “sense (see [21]) and D(2)φ should be understood in Aleksandrov sense, that is, as the absolutely continuous part of the distributional Hessian of the convex function φ; we denote it by D(2) Aφ.”
Optimal transportation, topology and uniqueness (2010, background): “The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42, 72,102], as well as applications in physics [38,75,97],”
Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates (2003, background): “It is known since the work of McCann [25] that U is displacement convex whenever U satisfies the dilation condition (1.”
A Sharp Stability Result for the Relative Isoperimetric Inequality Inside Convex Cones (2012, background): “mal transport theory, it is well known that there exists an a.e. unique map T: E!K(which we call the Brenier map) such that T= r˚where ˚is convex, T 2BV(Rn;K);and det(rT(x)) = 1 for a.e. x2E(see e.g. [5, 30, 29, 1]). Moreover, since T is the gradient of a convex function with positive Jacobian, rT(x”
Chapter 1 – Gradient Flows of Probability Measures (2007, background): “Section 3 is thus devoted to the analysis of convex functionals in P2(R) and to some particularly important examples, discovered by McCann [67].”
Gradient flow structures for discrete porous medium equations (2012, background): “In the continous setting, it follows from a fundamental result by McCann [12] that the Rényi entropy Fm(ρ) = 1m−1 ∫ Rn ρm(x) dx is displacement convex, i.e., convex along L2-Wasserstein geodesics in P(Rn), for m ≥ 1− 1n .”
Absolute continuity and summability of transport densities: simpler proofs and new estimates (2009, background): “Actually, the only bibliographical references which present specific useful tools for the proofs are [1], [2], [4] and [6], while we refer to [19], [18] and [12] for the general references on transport, displacement convexity and Γ−convergence; [5, 13, 14, 15, 16] are the main references for transpo”
Wasserstein Regression (2020, background): “Wasserstein Regression for t ∈ [0, 1], which is also referred to as McCann’s interpolation (McCann 1997).”