Convexity of the support of the displacement interpolation: Counterexamples

Abstract Given two smooth and positive densities ρ 0 , ρ 1 on two compact convex sets K 0 , K 1 , respectively, we consider the question whether the support of the measure ρ t obtained as the geodesic interpolant of ρ 0 and ρ 1 in the Wasserstein space W 2 ( R d ) is necessarily convex or not. We prove that this is not the case, even when ρ 0 and ρ 1 are uniform measures.

[1]  Nicola Gigli,et al.  On the inverse implication of Brenier-Mccann theorems and the structure of (P 2 (M),W 2 ) , 2011 .

[2]  Luis A. Caffarelli,et al.  A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity , 1990 .

[3]  Filippo Santambrogio,et al.  Optimal Transport for Applied Mathematicians , 2015 .

[4]  Guillaume Carlier,et al.  Barycenters in the Wasserstein Space , 2011, SIAM J. Math. Anal..

[5]  Alessio Figalli,et al.  The Monge–Ampère equation and its link to optimal transportation , 2013, 1310.6167.

[6]  Yann Brenier,et al.  A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem , 2000, Numerische Mathematik.

[7]  Filippo Santambrogio,et al.  Existence and Uniqueness of Equilibrium for a Spatial Model of Social Interactions , 2016 .

[8]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[9]  W. Gangbo,et al.  Optimal maps for the multidimensional Monge-Kantorovich problem , 1998 .

[10]  C. Villani.  The founding fathers of optimal transport , 2009 .

[11]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[12]  F. Santambrogio.  Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling , 2015 .

[13]  L. Caffarelli.  Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation , 1990 .

[14]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[15]  L. Caffarelli.  Some regularity properties of solutions of Monge Ampère equation , 1991 .