论文标题
半分化最佳传输几何形状的定量稳定性
Quantitative stability in the geometry of semi-discrete optimal transport
论文作者
论文摘要
我们显示了在半污垢最佳运输问题中产生的几何“细胞”的定量稳定性结果。我们的结果显示了两种类型的稳定性，第一个是相关的Laguerre细胞的稳定性，而没有任何有关源度量的连接性或规律性假设。第二个是在Hausdorff度量中的稳定性，在Poincar {è} - wIrtinger的不平等和规律性假设下，等于Monge-amp {è}的Ma-trudinger-wang条件等效。最后的结果还可以在双重电位函数的均匀规范中产生稳定性，所有三个稳定结果都具有明确的定量界限。我们的方法利用了图理论，凸几何形状和Monge-amp {è}的结合。
We show quantitative stability results for the geometric "cells" arising in semi-discrete optimal transport problems. Our results show two types of stability, the first is stability of the associated Laguerre cells in measure, without any connectedness or regularity assumptions on the source measure. The second is stability in Hausdorff measure, under a Poincar{è}-Wirtinger inequality and a regularity assumption equivalent to the Ma-Trudinger-Wang conditions of regularity in Monge-Amp{è}re. This last result also yields stability in the uniform norm of the dual potential functions, all three stability results come with explicit quantitative bounds. Our methods utilize a combination of graph theory, convex geometry, and Monge-Amp{è}re regularity theory.
