arXiv subjects
Publications and source records attributed to Jérôme Bertrand.
In this paper we provide conditions for a Wasserstein barycenter to be absolutely continuous with respect to its Hausdorff measure away from the vertices of the graph. One assumption removed in v2, the result is now optimal.
In this note, we solve the Gauss image problem given two Borel measures on the unit sphere, one of which is absolutely continuous with respect to the uniform measure.
We study stability of the spectral gap and observable diameter for metricmeasure spaces satisfying the RCD(1, $\infty$) condition. We show that if such a space has an almost maximal spectral gap, then it almost contains a Gaussian component, and the Laplacian has eigenvalues that are close to any integers, with dimension-free quantitative bounds. Under the additional assumption that the space admits a needle disintegration, we show that the spectral gap is almost maximal iff the observable diameter is almost maximal, again with quantitative dimension-free bounds.
Given a metric space X, one defines its Wasserstein space W2(X) as a set of sufficiently decaying probability measures on X endowed with a metric defined from optimal transportation. In this article, we continue the geometric study of W2(X) when X is a simply connected, nonpositively curved metric spaces by considering its isometry group. When X is Euclidean, the second named author proved that this isometry group is larger than the isometry group of X. In contrast, we prove here a rigidity result: when X is negatively curved, any isometry of W2(X) comes from an isometry of X.
The Gauss curvature measure of a pointed Euclidean convex body is a measure on the unit sphere which extends the notion of Gauss curvature to non-smooth bodies. Alexandrov's problem consists in finding a convex body with given curvature measure. In Euclidean space, A.D. Alexandrov gave a necessary and sufficient condition on the measure for this problem to have a solution. In this paper, we address Alexandrov's problem for convex bodies in the hyperbolic space H m+1. After defining the Gauss curvature measure of an arbitrary hyperbolic convex body, we completely solve Alexandrov's problem in this setting. Contrary to the Euclidean case, we also prove the uniqueness of such a convex body. The methods for proving existence and uniqueness of the solution to this problem are both new.
In this paper, we extend the DC Calculus introduced by Perelman on finite dimensional Alexandrov spaces with curvature bounded below. Among other things, our results allow us to define the Hessian and the Laplacian of DC functions (including distance functions as a particular instance) as a measure-valued tensor and a Radon measure respectively. We show that these objects share various properties with their analogues on smooth Riemannian manifolds.
In this paper we give a new proof of a theorem by Alexandrov on the Gauss curvature prescription of Euclidean convex sets. This proof is based on the duality theory of convex sets and on optimal mass transport. A noteworthy property of this proof is that it does not rely neither on the theory of convex polyhedra nor on P.D.E. methods (which appeared in all the previous proofs of this result).
We extend the geometric study of the Wasserstein space W(X) of a simply connected, negatively curved metric space X by investigating which pairs of boundary points can be linked by a geodesic, when X is a tree.
Optimal transport enables one to construct a metric on the set of (sufficiently small at infinity) probability measures on any (not too wild) metric space X, called its Wasserstein space W(X). In this paper we investigate the geometry of W(X) when X is a Hadamard space, by which we mean that $X$ has globally non-positive sectional curvature and is locally compact. Although it is known that -except in the case of the line- W(X) is not non-positively curved, our results show that W(X) have large-scale properties reminiscent of that of X. In particular we define a geodesic boundary for W(X) that enables us to prove a non-embeddablity result: if X has the visibility property, then the Euclidean plane does not admit any isometric embedding in W(X).