PreprintarXiv licencearXiv
On plank coverings of cube annuli
Abstract
We study plank coverings of cube annuli Q∖δQ⊂ℝd, where Q is the d-dimensional unit cube and δQ is a homothetic copy lying in its interior. A. Bezdek proved that, in dimension $2$, every plank covering of such an annulus has total width at least~1 whenever δ≤1−1/√2. Smurov, Bogataya, and Bogatyi extended this result to all d≥2 under the condition δ≤1−1/√d. It is conjectured that the same lower bound holds for δ≤1−1/d. We provide evidence for this conjecture in dimensions $2$ and $3$ by proving that there exist positive constants ε2 and ε3 such that the bound remains valid whenever δ≤1−1/√d+εd.
Related
BibTeX
@misc{ambrus2026plank,
title = {{On plank coverings of cube annuli}},
author = {Gergely Ambrus and Matthew Corbo and Dakota Nobuyuki Goto and Desmond Leitz and Kai Mawhinney},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18770},
eprint = {2610.18770},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18770},
}
RIS
TY - UNPB
TI - On plank coverings of cube annuli
AU - Gergely Ambrus
AU - Matthew Corbo
AU - Dakota Nobuyuki Goto
AU - Desmond Leitz
AU - Kai Mawhinney
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18770
UR - https://arxiv.org/abs/2610.18770
AB - We study plank coverings of cube annuli Q∖δQ⊂ℝ^d, where Q is the d-dimensional unit cube and δQ is a homothetic copy lying in its interior. A. Bezdek proved that, in dimension $2$, every plank covering of such an annulus has total width at least~1 whenever δ≤1−1/√2. Smurov, Bogataya, and Bogatyi extended this result to all d≥2 under the condition δ≤1−1/√d. It is conjectured that the same lower bound holds for δ≤1−1/d. We provide evidence for this conjecture in dimensions $2$ and $3$ by proving that there exist positive constants ε_2 and ε_3 such that the bound remains valid whenever δ≤1−1/√d+ε_d.
ER -