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

Gregoire Loeper

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arXiv · 1301.6229

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

Gregoire Loeper

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Abstract

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.

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Gregoire Loeper. 2013-01-26. Regularity of optimal maps on the sphere: the quadratic cost and the reflector antenna. https://arxiv.org/abs/1301.6229

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