The View-Obstruction Problem for Spheres in Dimensions Six and Seven
Abstract
Place closed Euclidean balls of radius r at the points of (1/2, 1/2, ..., 1/2) + Z^n. The view-obstruction problem for spheres asks for the least r such that every ray from the origin into the open positive orthant meets a ball. Equivalently, it asks for D_n = sup_v min_{t in R} sum_i ||t v_i - 1/2||{Z}^2 over positive integer vectors v, where ||x||{Z} denotes the distance from x to the nearest integer. The values for n <= 5 are classical. We prove D_6 = 107/312, attained only by permutations of (12, 9, 7, 5, 3, 2), and D_7 = 73/180, attained only by permutations of (11, 10, 8, 7, 4, 3, 1). The proof reduces the supremum to a finite search. The key facts are an exact lower bound d_n(v) >= delta^2_U for every rational subspace U perpendicular to v, a matching fiber upper bound, and Banaszczyk's transference theorem, applied along the Korkine-Zolotarev flag of the dual lattice v^perpendicular intersection Z^n. The search is quotiented by signed coordinate permutations and carried out entirely in exact rational arithmetic. It terminates with a certificate: no node of the search is left unresolved, and every surviving candidate is evaluated exactly. The method also recovers the known values for n = 3, 4, 5.
Related
- Quadratic Scalar Curvature Decay and Uniform Positivity in Dimensions Four through Seven
- Borsuk number in four dimensions is at most seven
- The recognition problem for 4-manifolds and the unknotting problem for 2-spheres in 4-manifolds
- Positive sectional curvature on exotic spheres in infinitely many dimensions
- A Computational Obstruction to Swapping Area and Dinv: An Automata-Theoretic View of the q,t-Catalan Symmetry
BibTeX
@misc{kahlon2026viewobstruction,
title = {{The View-Obstruction Problem for Spheres in Dimensions Six and Seven}},
author = {Vineet Kahlon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37711},
eprint = {2610.37711},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37711},
}
RIS
TY - UNPB
TI - The View-Obstruction Problem for Spheres in Dimensions Six and Seven
AU - Vineet Kahlon
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.37711
UR - https://arxiv.org/abs/2610.37711
AB - Place closed Euclidean balls of radius r at the points of (1/2, 1/2, ..., 1/2) + Z^n. The view-obstruction problem for spheres asks for the least r such that every ray from the origin into the open positive orthant meets a ball. Equivalently, it asks for D_n = sup_v min_{t in R} sum_i ||t v_i - 1/2||{Z}^2 over positive integer vectors v, where ||x||{Z} denotes the distance from x to the nearest integer. The values for n <= 5 are classical. We prove D_6 = 107/312, attained only by permutations of (12, 9, 7, 5, 3, 2), and D_7 = 73/180, attained only by permutations of (11, 10, 8, 7, 4, 3, 1). The proof reduces the supremum to a finite search. The key facts are an exact lower bound d_n(v) >= delta^2_U for every rational subspace U perpendicular to v, a matching fiber upper bound, and Banaszczyk's transference theorem, applied along the Korkine-Zolotarev flag of the dual lattice v^perpendicular intersection Z^n. The search is quotiented by signed coordinate permutations and carried out entirely in exact rational arithmetic. It terminates with a certificate: no node of the search is left unresolved, and every surviving candidate is evaluated exactly. The method also recovers the known values for n = 3, 4, 5.
ER -