Non-dissective coverings by planks

2025-11-25

AI Summary

This paper investigates the non-partitioned plank covering problem for the unit sphere: given planks of equal width ε (regions between two parallel hyperplanes), can they be sequentially translated and placed such that the uncovered region remains connected after each placement, ultimately achieving full coverage? Addressing Groemer’s classical result, we establish the first nontrivial upper and lower bounds for this connectivity-constrained setting: an upper bound of (C/varepsilon^{7/4}) and a lower bound of (c/varepsilon^{4/3}), revealing the fundamental limits on covering efficiency under connectivity preservation. Methodologically, we integrate geometric probability with convex covering theory, designing a computationally efficient greedy algorithm that maintains connectivity of the uncovered set throughout the incremental covering process. Our key contributions are threefold: (i) the first quantitative characterization of the compactness bottleneck in non-partitioned spherical covering; (ii) the first rigorous proof of a nontrivial lower bound; and (iii) an explicit, constructive upper bound via a feasible covering scheme.

Technology Category

• Planning, Routing, and Scheduling: Planning under Uncertainty
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

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 $C/ε^{7/4}$ 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 $c/ε^{4/3}$ for this quantity.

Problem

• Solving converse of Tarski's plank problem about covering unit balls
• Establishing sufficient conditions for non-dissective translative plank coverings
• Providing algorithmic proof with complexity bounds for coverage problem

Innovation

• Non-dissective translative covering algorithm for planks
• Improved upper bound of C/ε^(7/4) for plank covering
• Established lower bound of c/ε^(4/3) for covering complexity