标题： 流形上弗雷歇均值的切割点的零概率 显示英文标题

标题： Zero Probability of the Cut Locus of a Fréchet Mean on a Riemannian Manifold

摘要： 我们证明了在连通且完备的黎曼流形上，随机变量的弗雷歇均值的切割点集具有零概率，这一结果在特殊情况下已知，并在一般情况下被猜想过。 在应用中，我们排除了粘性，同时提供了处处不光滑的弗雷歇函数的例子，并讨论了该结论到弗雷歇$p$-均值的扩展，其中$p\neq 2$，以及到非完备流形和更一般的度量空间的情况。 显示英文摘要

摘要： We show that the cut locus of a Fr\'echet mean of a random variable on a connected and complete Riemanian manifold has zero probability, a result known previously in special cases and conjectured in general. In application, we rule out stickiness, while providing examples of nowhere smooth Fr\'echet functions and we discuss extensions of the statement to Fr\'echet $p$-means, for $p\neq 2$, as well as to noncomplete manifolds and more general metric spaces.