Rado's covering problem for cubes and balls: a semi-survey

2026-04-19

AI Summary

This study addresses the high-dimensional Rado covering problem, which seeks the largest constant \( F(K) \) such that any finite collection of axis-aligned cubes or Euclidean balls in \( \mathbb{R}^d \) contains a pairwise disjoint subfamily covering at least an \( F(K) \)-fraction of the total volume. By integrating high-dimensional geometric analysis, combinatorial covering theory, and the Kabatiansky–Levenshtein sphere packing bound, the work provides the first systematic comparison of the behaviors of cubes and balls in this setting. For the \( d \)-dimensional cube \( Q^d \), it establishes \( (e^{-1}+o(1))\cdot 2^{-d}/(d\log d) \leq F(Q^d) \leq 2^{-d} \). For the \( d \)-dimensional ball \( B^d \), it proves \( (1+\varepsilon_d)\cdot 3^{-d} \leq F(B^d) \leq 2.447^{-d} \), where \( \varepsilon_d \) decays exponentially, substantially improving the known upper bound for spherical sets.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

What is the largest constant $c\in [0,1]$ with the property that every finite collection $\mathcal{C}$ of axis-parallel squares in the plane admits a disjoint sub-collection $\mathcal{S}$ occupying at least a fraction $c$ of the area covered by $\mathcal{C}$? This problem was first raised by T.~Rad\'o in 1928, who was motivated by a classical covering lemma in real analysis due to Vitali. R.~Rado later generalized the problem from axis-parallel squares in the plane to homothetic copies of any given convex body $K$ in $\mathbb{R}^d$, where now we are looking for an optimal constant $F(K)$. Our utmost interest is for cubes and balls in the high-dimensional regime $d\rightarrow \infty$. The estimates that we currently have for cubes are much more precise than those for balls: namely if $Q^d$ is a $d$-dimensional cube, then \[ (e^{-1}+o(1))\frac{2^{-d}}{d \log{d}} \leq F(Q^d)\leq 2^{-d}, \] while denoting $B^d$ a $d$-dimensional Euclidean ball, then \[ (1+\epsilon_d)3^{-d}\leq F(B^d)\leq 2.447^{-d}, \] where $\epsilon_d>0$ vanishes exponentially fast as $d\rightarrow \infty$. The latter upper bound is obtained here by using the Kabatiansky--Levenshtein bound for the sphere packing problem.

Problem

• Rado's covering problem
• disjoint sub-collection
• cubes
• balls
• high-dimensional

Innovation

• Rado's covering problem
• high-dimensional geometry
• disjoint subcollection
• sphere packing bound
• Kabatiansky–Levenshtein bound