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Robert McCann obtained, generalizing results for the Euclidean case by Yann Brenier, the polar factorization of Borel maps S : M -> M pushing forward $\\mu$ to a measure $\\nu$: each S factors uniquely a.e. into the composition S = T \\circ U, where U : M -> M is volume preserving and T : M -> M is the optimal map transporting $\\mu$ to $\\nu$ with respect to the cost function d^2/2.\n  In this article we study the polar factorization of conformal and projective maps of the sphere S^n. 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