Dense uniqueness and dense nonuniqueness of Fréchet means
Abstract
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 < ∞, also.
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BibTeX
@misc{huckemann2026dense,
title = {{Dense uniqueness and dense nonuniqueness of Fréchet means}},
author = {Stephan F. Huckemann and Alexander Lytchak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11946},
eprint = {2610.11946},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11946},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dense uniqueness and dense nonuniqueness of Fréchet means
AU - Stephan F. Huckemann
AU - Alexander Lytchak
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11946
UR - https://arxiv.org/abs/2610.11946
AB - 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 < ∞, also.
ER -