REVIEW 1 major objections 1 cited by
A problem of intersection of balls in normed space
T0 review · 1 major / 0 minor · reviewed 2026-06-29 · grok-4.3
Pith's one-line read In any normed plane, the intersection of small open balls minus a large closed ball is contractible if nonempty.
desk verdict The paper proves that in any normed plane a nonempty set formed by intersecting small open balls and removing one large closed ball is contractible. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The set formed by intersecting finitely many small open balls and subtracting one large closed ball, whose contractibility is shown in two-dimensional normed spaces.
What would settle it
An explicit two-dimensional normed space together with concrete balls whose intersection minus the large ball is nonempty yet fails to be contractible.
Extended reading notes
Core claim
It is proved that in an arbitrary normed plane, the set obtained by removing a large closed ball from a finite intersection of small open balls is always contractible, provided that it is non-empty.
Load-bearing premise
The ambient space must be exactly two-dimensional.
Editorial extensions
If this is right
- The sets can be used in covers without introducing holes or higher-dimensional topological features.
- The contractibility holds for every possible norm on the plane, not just the Euclidean one.
- No additional assumptions on the radii or centers are required beyond the set being nonempty.
- The result supplies a topological guarantee for constructions that appear in Gromov-Hausdorff distance estimates.
Reading between the lines
- The same contractibility may fail in dimensions three and higher, requiring extra conditions.
- The proof technique might adapt to show simple connectedness or vanishing of other invariants in related metric settings.
- Explicit deformation retractions could be constructed algorithmically for computational use in low dimensions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript investigates topological properties of intersections of balls in finite-dimensional normed spaces, motivated by applications to Gromov-Hausdorff distance estimation. It claims to prove that in an arbitrary normed plane, the set obtained by removing a large closed ball from a finite intersection of small open balls is always contractible whenever the set is non-empty.
Significance. If rigorously established, the result would supply a concrete topological fact about contractibility for a specific class of sets (intersections of convex balls minus a ball) in 2-dimensional normed spaces. This could support constructions of covers in metric geometry, but the dimensional restriction to planes and the convex nature of the sets make the claim plausible rather than surprising.
major comments (1)
- [Abstract] Abstract: the statement asserts that 'It is proved that...' the indicated set is contractible, yet supplies no derivation outline, lemmas, key steps, or verification that the topological argument holds in an arbitrary norm; this gap is load-bearing for the central claim and prevents assessment of soundness.
Simulated Author's Rebuttal
We thank the referee for their comments. We address the major comment point by point below.
read point-by-point responses
-
Referee: [Abstract] Abstract: the statement asserts that 'It is proved that...' the indicated set is contractible, yet supplies no derivation outline, lemmas, key steps, or verification that the topological argument holds in an arbitrary norm; this gap is load-bearing for the central claim and prevents assessment of soundness.
Authors: The abstract is intentionally concise, as is standard, but we agree it provides no outline of the argument. The full manuscript develops the proof in detail, including the key steps: first establishing that the intersection of small open balls is convex and open, then constructing an explicit deformation retraction onto a point within the complement of the large closed ball using the triangle inequality in the arbitrary norm and radial projection from a suitable interior point. This works specifically in 2D due to the topology of the plane. To improve clarity, we will revise the abstract to include a one-sentence outline of this strategy. revision: yes
Circularity Check
No significant circularity
full rationale
The paper states a direct topological result: the indicated intersection of balls (after removing a large closed ball) is contractible in any 2-dimensional normed space when non-empty. No equations, parameters, or premises reduce to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation relies on standard convexity and contractibility arguments in finite-dimensional normed spaces without importing uniqueness theorems or ansatzes from prior author work. The claim is dimensionally restricted by design and self-contained against external topological benchmarks.
Assumptions & free parameters
assumptions (2)
- standard math Normed spaces are vector spaces equipped with a norm that induces a metric and defines open and closed balls.
- standard math Contractibility is a topological property that can be verified via continuous deformations in finite-dimensional spaces.
Cite this review
Pith. "Pith review of A problem of intersection of balls in normed space." pith.science (2026). https://pith.science/paper/YT6YOIW4
@misc{pith2026260627583,
author = {Pith},
title = {Pith review of: A problem of intersection of balls in normed space},
year = {2026},
howpublished = {\url{https://pith.science/paper/YT6YOIW4}},
note = {Machine review of arXiv:2606.27583}
}
read the original abstract
This paper investigates the topological properties of intersections of balls in finite-dimensional normed spaces - a problem that naturally arises when constructing covers for estimating the Gromov-Hausdorff distance. We study the topology of a set obtained by removing a large closed ball from a finite intersection of small open balls. It is proved that in an arbitrary normed plane, such a set is always contractible, provided that it is non-empty.
Forward citations
Cited by 1 Pith paper
-
Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces
The paper gives a Gromov–Hausdorff lower bound via the relative Jung constant (sound) but its stronger Jung-constant version relies on an impossible normalization step.
Reference graph
Works this paper leans on
-
[1]
Adams, F
H. Adams, F. Frick, S. Majhi, and N. McBride. Hausdorff vs Gromov–Hausdorff distances. Discrete and Computational Geometry, 2025
2025
-
[2]
Burago, Yu
D. Burago, Yu. Burago, S. Ivanov, A Course in Metric Geometry, Graduate Studies in Math- ematics 33 AMS, Providence, RI, 2001
2001
-
[3]
Fomenko and D
A. Fomenko and D. Fuchs. Homotopical Topology. Moscow University Press. 2016
2016
-
[4]
A. Hatcher. Algebraic Topology. Cambridge University Press. 2002. 7
2002
-
[5]
A.O. Ivanov, I.N. Mikhailov, and A.A. Tuzhilin. Gromov-Hausdorff Geometry of Metric Trees em ArXiv e-prints, 2024, arXiv:2412.18888 [math.MG]
-
[6]
E. S. Polovinkin, M. V. Balashov. Elements of convex and strongly convex analysis. Moscow: Fizmatlit. 2007
2007
-
[7]
Who Invented the Gromov-Hausdorff Distance?
A. Tuzhilin. Who Invented the Gromov-Hausdorff Distance? ArXiv e-prints, arXiv:1612.00728, 2016. 8
work page Pith review arXiv 2016
Reviewed June 29, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.