Six-point persistence on the Euclidean two-sphere
Give the unit sphere $S^2 \subset \mathbb{R}^3$ the chordal (Euclidean) distance. For a six-point set $X \subset S^2$ and $x \in X$, let $r_1(x) \ge r_2(x)$ be the largest and second-largest of the five distances from $x$ to the other points, counting ties, and put $$b(X) = \max_{x \in X} r_2(x), \qquad d(X) = \min_{x \in X} r_1(x),$$ $$\mathcal{P} = \{(b(X), d(X)) : X \subset S^2,\ |X| = 6,\ b(X) < d(X)\}.$$ Gómez and Mémoli showed that the degree-two Vietoris–Rips persistence diagram of a six-point metric space $X$ is the single point $(b(X), d(X))$ when $b(X) < d(X)$ and is empty otherwise. So $\mathcal{P}$ is the set of nonzero degree-two persistence pairs of samples of at most six points of $S^2$, in the Vietoris–Rips filtration with simplices of diameter at most $t$ and coefficients in any field; samples of fewer than six points contribute none.
Problem. Is $$\mathcal{P} = \Bigl\{(b,d) : 0 < b < d \le 2,\ \ d \le \frac{2b}{\sqrt{3}} \ \text{ or } \ \Bigl(b \ge \sqrt{2} \ \text{ and } \ \frac{4b^2(3 - b^2)}{4 - b^2} \le d^2\Bigr)\Bigr\}?$$
Source: Gómez and Mémoli (2024), Conjecture 5.25, for the region of their Proposition 5.24, with the condition b ≥ √2 that its proof imposes on the second branch and the printed formula omits; AIM — Open Applied Problems states the printed region, which is refuted by d ≤ √2·b.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- AT
- Keywords
- AIM Open Applied Problems, persistence sets, two-sphere, Vietoris–Rips persistence
References
@article{gomez2024curvature,
author = {G{\'o}mez, Mario and M{\'e}moli, Facundo},
title = {Curvature sets over persistence diagrams},
journal = {Discrete Comput. Geom.},
volume = {72},
number = {1},
year = {2024},
pages = {91--180},
doi = {10.1007/s00454-024-00634-0},
eprint = {2103.04470},
archivePrefix = {arXiv},
}
@misc{aim-six-point-sphere-persistence,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Six-point persistence on the Euclidean two-sphere}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/521-six-point-sphere-persistence.md},
note = {Entry 521 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00843, "Six-point persistence on the Euclidean two-sphere", clopen.solutions.