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MR2569074 49Q20 (58J65 60B99)
Figalli, Alessio; Rifford, Ludovic Continuity of optimal transport maps and convexity of injectivity domains on small deformations of $\Bbb S^2$. Comm. Pure Appl. Math. 62 (2009), no. 12, 1670–1706.
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