Critical Diameters and Vietoris-Rips Filtrations
Abstract
We relate critical diameters of finite configurations to the topology of Vietoris-Rips complexes. For a compact metric space and 0<r<s, we prove that the canonical inclusion from scale r to scale s is a homotopy equivalence whenever [r,s) contains no diameter of a finite labelled configuration at which the diameter function has zero weak slope. On Riemannian manifolds, weak-slope stationarity implies Clarke criticality. At positive diameter, when the distances realizing the diameter are smooth, both are equivalent to first-order stationarity: the absence of a direction decreasing all these distances to first order. For closed connected smooth manifolds, the Clarke critical diameter spectrum has Hausdorff dimension zero, even at the cut locus. In the real-analytic case, it is finite at each fixed number of labels and countable over all label numbers. On the unit round sphere Sm, m≥1, positive nonantipodal weak-slope stationary configurations are characterized by nonzero nonnegative equilibrium stresses. The least positive critical diameter is arccos(−1/(m+1)), and the first accumulation point is arccos(−1/m). We lift every such stress through a spherical stack construction adapted from Lovász, producing stationary diameters that approach the original value from below in the next dimension as the number of layers increases. We also give example in general metric space showing that weak-slope stationary diameters need not correspond to changes in the homotopy type of Vietoris-Rips complexes.
Related
- Homotopy distributions for Vietoris-Rips complexes of random circle samples
- Vietoris-Rips complexes of affine Coxeter groups
- Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds
- Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces
- Cayley graph diameters for fixed cycle types are eventually quasipolynomial
BibTeX
@misc{memoli2026critical,
title = {{Critical Diameters and Vietoris-Rips Filtrations}},
author = {Facundo Mémoli and Qingsong Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.33009},
eprint = {2609.33009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33009},
}
RIS
TY - UNPB
TI - Critical Diameters and Vietoris-Rips Filtrations
AU - Facundo Mémoli
AU - Qingsong Wang
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2609.33009
UR - https://arxiv.org/abs/2609.33009
AB - We relate critical diameters of finite configurations to the topology of Vietoris-Rips complexes. For a compact metric space and 0<r<s, we prove that the canonical inclusion from scale r to scale s is a homotopy equivalence whenever [r,s) contains no diameter of a finite labelled configuration at which the diameter function has zero weak slope. On Riemannian manifolds, weak-slope stationarity implies Clarke criticality. At positive diameter, when the distances realizing the diameter are smooth, both are equivalent to first-order stationarity: the absence of a direction decreasing all these distances to first order. For closed connected smooth manifolds, the Clarke critical diameter spectrum has Hausdorff dimension zero, even at the cut locus. In the real-analytic case, it is finite at each fixed number of labels and countable over all label numbers. On the unit round sphere S^m, m≥1, positive nonantipodal weak-slope stationary configurations are characterized by nonzero nonnegative equilibrium stresses. The least positive critical diameter is arccos(−1/(m+1)), and the first accumulation point is arccos(−1/m). We lift every such stress through a spherical stack construction adapted from Lovász, producing stationary diameters that approach the original value from below in the next dimension as the number of layers increases. We also give example in general metric space showing that weak-slope stationary diameters need not correspond to changes in the homotopy type of Vietoris-Rips complexes.
ER -