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Non-Dissective Coverings by Planks

Authors Andrey Kupavskii , János Pach

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Tarski’s plank problem
  • translative cover
  • non-dissective cover

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Abstract

A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viewed as the converse of Tarski’s plank problem (Bang’s theorem): Is it true that if the total width of a collection of planks is sufficiently large, then the planks can be individually translated to cover a unit ball B?
A translative covering of B by planks is said to be non-dissective if the planks can be added one by one, in some order, such that the uncovered part remains connected at each step and is empty at the end. Improving a classical result of Groemer, we show that every set of C/ε^{7/4} planks of width ε admits a non-dissective translative covering of a 3-dimensional ball B³, provided C is large enough. Our proof yields a low-complexity algorithm. We also show that c/ε^{4/3} planks are, in general, insufficient for a non-dissective covering of B³. This provides the first non-trivial lower bound for this problem.

Cite As Get BibTex

Andrey Kupavskii and János Pach. Non-Dissective Coverings by Planks. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 67:1-67:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.67

Author Details

Andrey Kupavskii
  • Moscow Institute of Physics and Technology, Russia
  • Saint-Petersburg University, Russia
  • Innopolis University, Tatarstan Republic, Russia
János Pach
  • Alfréd Rényi Institute of Mathematics, Hungary

Funding

  • Kupavskii, Andrey: Supported by the Ministry of Science and Higher Education of the Russian Federation, project No. FSMG-2024-0025.
  • Pach, János: Supported by NKFIH grant K-131529 and ERC Advanced Grant 882971 "GeoScape."

References

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