Building on the results of Ma, Trudinger and Wang \cite{MTW}, and of the
author \cite{L5}, we study two problems of optimal transportation on the
sphere: the first corresponds to the cost function $d^2(x,y)$, where
$d(\cdot,\cdot)$ is the Riemannian distance of the round sphere; the second
corresponds to the cost function $-\log|x-y|$, it is known as the reflector
antenna problem. We show that in both cases, the {\em cost-sectional curvature}
is uniformly positive, and establish the geometrical properties so that the
results of \cite{L5} and \cite{MTW} can apply: global smooth solutions exist
for arbitrary smooth positive data and optimal maps are H\"older continuous
under weak assumptions on the data.

本文研究了在球面上的两个最优输运问题，证明了成本切面曲率是一致正的，并建立了几何性质，证明了全局平滑解是存在的，最优映射在数据的弱假设下是 Hölder 连续的。

球面上最优映射的正则性：二次代价与反射天线

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

Among $\R^3$-valued triples of random vectors $(X,Y,Z)$ having fixed marginal
probability laws, what is the best way to jointly draw $(X,Y,Z)$ in such a way
that the simplex generated by $(X,Y,Z)$ has maximal average volume? Motivated
by this simple question, we study optimal transportation problems with several
marginals when the objective function is the determinant or its absolute value.

研究固定边际概率律的三元随机向量 $(X,Y,Z)$ 的最佳绘制方式，使得由 $(X,Y,Z)$ 生成的单纯形具有最大平均体积的最优输运问题，其中目标函数为行列式或其绝对值。