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A sharp plank theorem for cube annuli
Abstract
We prove a conjecture of András Bezdek on plank coverings of square annuli. A plank in a Euclidean space is the closed region between two distinct parallel hyperplanes, and its width is the distance between them. Let Q be a unit square and let H be a square of side length 1/2 whose sides are parallel to those of Q. Then every finite family of planks covering Q∖ H has total width at least $1$. More generally, if an axis-parallel box with side lengths h1,…,hd is removed from the unit cube in ℝd and ∑jhj = d−1, then every finite plank covering of the remaining set has total width at least one. The bounds are sharp.
Related
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- The sharp exponent for the Banach--Mazur distance to the cube
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BibTeX
@misc{bakaev2026sharp,
title = {{A sharp plank theorem for cube annuli}},
author = {Egor Bakaev and Alexander Polyanskii},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00379},
eprint = {2611.00379},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00379},
}
RIS
TY - UNPB
TI - A sharp plank theorem for cube annuli
AU - Egor Bakaev
AU - Alexander Polyanskii
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00379
UR - https://arxiv.org/abs/2611.00379
AB - We prove a conjecture of András Bezdek on plank coverings of square annuli. A plank in a Euclidean space is the closed region between two distinct parallel hyperplanes, and its width is the distance between them. Let Q be a unit square and let H be a square of side length 1/2 whose sides are parallel to those of Q. Then every finite family of planks covering Q∖ H has total width at least $1$. More generally, if an axis-parallel box with side lengths h_1,…,h_d is removed from the unit cube in ℝ^d and ∑_jh_j = d−1, then every finite plank covering of the remaining set has total width at least one. The bounds are sharp.
ER -