球上的质量输运泛函及其在对数Minkowski问题中的应用
Mass transportation functionals on the sphere with applications to the logarithmic Minkowski problem
摘要：
本文研究了单位球$S^{n-1}$上的运输问题，给出了对称概率测度和费用函数$c(x,y) = \log \frac{1}{\langle x, y \rangle}$。我们计算了相应的Kantorovich泛函$K$的变分，并研究了$S^{n-1}$上的一个自然关联的度量测度空间，该度量测度空间被赋予了由相应的输运势产生的黎曼度量.我们引入了一个新的运输功能的最小值是解决对称的log-Minkowski问题，并证明了$K$满足以下类似的高斯运输不等式的均匀概率测度$S^{n-1}$：$\frac{1}{n} Ent(ν) \ge K(σ, ν)$。结果表明，我们的结果与欧几里得空间上的K{}hler-Einstein方程的理论有着显著的相似性.作为副产品，我们得到了一致测度的log-Minkowski问题解的唯一性的一个新的证明。
英文摘要：
We study the transportation problem on the unit sphere $S^{n-1}$ for symmetric probability measures and the cost function $c(x,y) = \log \frac{1}{\langle x, y \rangle}$. We calculate the variation of the corresponding Kantorovich functional $K$ and study a naturally associated metric-measure space on $S^{n-1}$ endowed with a Riemannian metric generated by the corresponding transportational potential. We introduce a new transportational functional which minimizers are solutions to the symmetric log-Minkowski problem and prove that $K$ satisfies the following analog of the Gaussian transportation inequality for the uniform probability measure $σ$ on $S^{n-1}$: $\frac{1}{n} Ent(ν) \ge K(σ, ν)$. It is shown that there exists a remarkable similarity between our results and the theory of the K{ä}hler-Einstein equation on Euclidean space. As a by-product we obtain a new proof of uniqueness of solution to the log-Minkowski problem for the uniform measure.
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