Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna

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.