2026-09-05T11:55:18Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:25874 2026-06-23T13:00:00Z ddc:004 open_access

Non-Dissective Coverings by Planks Kupavskii, Andrey Pach, János Tarski’s plank problem translative cover non-dissective cover 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Andrey Kupavskii and János Pach 2026 Is Part Of LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2026.67 urn:nbn:de:0030-drops-258743 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.67 eng https://creativecommons.org/licenses/by/4.0/legalcode