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Chao ChenLitan Yan 《Statistics & probability letters》2011,81(8):1003-1012
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Let A⊂Rd, d?2, be a compact convex set and let be a probability measure on A equivalent to the restriction of Lebesgue measure. Let be a probability measure on equivalent to the restriction of Lebesgue measure. We prove that there exists a mapping T such that ν=μ○T−1 and T=φ⋅n, where is a continuous potential with convex sub-level sets and n is the Gauss map of the corresponding level sets of φ. Moreover, T is invertible and essentially unique. Our proof employs the optimal transportation techniques. We show that in the case of smooth φ the level sets of φ are governed by the Gauss curvature flow , where K is the Gauss curvature. As a by-product one can reprove the existence of weak solutions to the classical Gauss curvature flow starting from a convex hypersurface. 相似文献
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