Regularity of Optimal Maps on the Sphere: the Quadratic Cost and the Reflector Antenna

By Grégoire Loeper. 2013

109 citations recorded by the source corpus.

Building on the results of Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151–183 (2005)), and of the author Loeper (in Acta Math., to appear), we study two problems of optimal transportation on the sphere: the first corresponds to the cost function d2(x, y), where d(·, ·) is the Riemannian distance of the round sphere; the second corresponds to the cost function −log |x − y|, known as the reflector antenna problem. We show that in both cases, the cost-sectional curvature is uniformly positive, and establish the geometrical properties so that the results of Loeper (in Acta Math., to appear) and Ma et al. (in Arch. Rational Mech. Anal. 177(2), 151–183 (2005)) can apply: global smooth solutions exist for arbitrary smooth positive data and optimal maps are Hölder continuous under weak assumptions on the data.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f5d553eb97f39643caf3b57af2f7a5738ea3602a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Regularity of optimal transport in curved geometry: The nonfocal case (2010, background): “Condition (1.2) implies that the sectional curvature of M is bounded below by K. Loeper [ 21 ] showed that the round sphere Sn satisfies (1.2) for some”
Wasserstein Barycenters over Riemannian manifolds (2014, background): “Nonnegatively cross curved spaces include, for example, Euclidean space, the round sphere [25] and its small perturbations in two dimensions [11, 15], products and quotients of these [20], and some other symmetric spaces [10], but exclude many other spaces, including any manifold with negative secti”
Partial regularity for optimal transport maps (2012, background): “Let us also mention that, up to now, the MTW condition is known to be satisfied only for very special classes of Riemannian manifolds, such as spheres, their products, their quotients and submersions, and their perturbations [10, 11, 20, 22, 25, 29, 32], and for instance it is known to fail on suffi”
The Monge–Ampère equation and its link to optimal transportation (2013, background): “Some examples of manifolds satisfying the MTW condition have been found in [55, 57, 73, 74, 84, 85]: • R and T satisfy MTW(0).”
Hölder Continuity and Injectivity of Optimal Maps (2011, background): “Examples include the Euclidean distance squared between two convex graphs over two sufficiently convex sets in Rn [45], the simple harmonic oscillator action [38], and the Riemannian distance squared on the following spaces: the round sphere [43] and perturbations thereof [13] [21] [22], multiple pr”