AI 中文总结
该研究证明在两类度量下，具有唯一和非唯一Fréchet均值的概率分布均稠密，且无开集存在连续均值选择，强化了Dirac嵌入零可达性结论，论证结合割迹规避与拓扑障碍，结论适用于曲率有下界的Alexandrov空间及Riemann流形。
AI 中文摘要
从已有结果可推得，在二次Wasserstein度量和全变差度量下，具有唯一Fréchet均值的概率分布，在所有存在均值的测度中是稠密的。此外，我们证明在曲率有下界的完备有限维非收缩Alexandrov空间（包含无边界紧致流形和非流形形状空间）上，逆命题也成立：具有非唯一Fréchet均值的概率分布，在上述两种度量下同样是稠密的。而且，在任一度量中，概率测度的任意非空开集都不存在连续的均值选择映射。这一结果强化了关于Dirac嵌入零可达性的已有结论。我们的论证结合了原子的割迹规避与拓扑障碍。对于Riemann流形，我们的所有结论对所有指数1 < p < ∞也成立。
英文摘要
From previous results it follows that probability distributions on metric spaces featuring unique Fréchet means are dense among measures admitting means in both the quadratic Wasserstein metric and the total variation metric. Additionally, we show that the converse also holds on complete finite-dimensional noncontractible Alexandrov spaces with curvatures bounded from below, which encompass compact manifolds without boundaries and nonmanifold shape spaces: Probability distributions featuring nonunique Fréchet means are also dense in both the Wasserstein metric and the total variation metric. Moreover, in either metric, no nonempty open set of probability measures admits a continuous selection of means. This sharpens a previous result on zero reach of the Dirac embedding. Our argument combines cut-locus avoidance for atoms with a topological obstruction. For Riemannian manifolds, all of our conclusions hold for all exponents $1 < p < \infty$, also.
