2025/11/25 by Kupavskii, Andrey, Pach, Janos
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Approximation and Integration #Point processes and geometric inequalities
paper · doi:10.48550/arxiv.2511.20047
openalex publication_date 2025/11/25 · openalex created_date 2025/11/28 · openalex updated_date 2026/07/28
A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed 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/ε7/4 planks of width ε admits a non-dissective translative covering of B, provided C is large enough. Our proof yields a low-complexity algorithm. We also establish the first nontrivial lower bound of c/ε4/3 for this quantity.