On the Regularity of Optimal Transportation Potentials on round Spheres
نویسنده
چکیده
In this paper the regularity of optimal transportation potentials dened on round spheres is investigated. Speci cally, this research generalises the calculations done by Loeper, where he showed that the strong (A3) condition of Trudinger and Wang is satis ed on the round sphere, when the cost-function is the geodesic distance squared. In order to generalise Loeper's calculation to a broader class of cost-functions, the (A3) condition is reformulated via a stereographic projection that maps charts of the sphere into Euclidean space. This reformulation subsequently allows one to verify the (A3) condition for any case where the cost-fuction of the associated optimal transportation problem can be expressed as a function of the geodesic distance between points on a round sphere. With this, several examples of such cost-functions are then analysed to see whether or not they satisfy this (A3) condition.
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