We consider the problem of identifying the worst point-symmetric shape for covering n-dimensional Euclidean space with lattice translates. Here we focus on the dimensions where the thinnest lattice covering with balls is known and ask whether the ball is a pessimum for covering in these dimensions compared to all point-symmetric convex shapes. We find that the ball is a local pessimum in 3 dimensions, but not so for 4 and 5 dimensions.
We note that the recent polynomial proofs of the spherical and complex plank covering problems by Zhao and Ortega-Moreno give some general information on zeros of real and complex polynomials restricted to the unit sphere. As a corollary of these results, we establish several generalizations of the celebrated Bang plank covering theorem. We prove a tight polynomial analog of the Bang theorem for the Euclidean ball and an even stronger polynomial version for the complex projective space. Specifically, for the ball we show that for every real nonzero $d$-variate polynomial $P$ of degree $n$, the…
We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vaxès in 2007 in the special case of planar graphs and of balls having the same radius.
Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric models, and with the discrete coupon collector's problem and with cover times for finite Markov chains, one expects a "weak concentration" bound for the distribution of the cover time to hold under minimal assumptions. We give two such results, one for random fixed-radius balls and the other for sequentially arriving randomly-centered and deterministically growing balls. Each is in fact a simple application of a different more general bound, the f…
Rado's covering problem for cubes and balls: a semi-survey
Istituto Nazionale di Alta Matematica “F. Severi”（高等数学国家研究所“F. Severi”）；Scuola Normale Superiore, Pisa（比萨高等师范学院）；Algoresearch L.L.C.（Algoreresearch有限责任公司）；Research Institute of the University of Bucharest（布加勒斯特大学研究所）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）
The present work surveys problems in $n$-dimensional space with $n$ large. Early development in the study of packing and covering in high dimensions was motivated by the geometry of numbers. Subsequent results, such as the discovery of the Leech lattice and the linear programming bound, which culminated in the recent solution of the sphere packing problem in dimensions 8 and 24, were influenced by coding theory. After mentioning the known results concerning existence of economical packings and coverings we discuss the different methods yielding upper bounds for the density of packing congruent…
On the weighted hard-core model and Rado's covering problem for congruent Euclidean balls
Institute of Mathematics and Interdisciplinary Sciences, Xidian University（西安电子科技大学数学与交叉科学研究院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
Covering large-dimensional Euclidean spaces by random translates of a given convex body
Carnegie Mellon University（卡内基梅隆大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Science and Technology of China（中国科学技术大学）；University of Warwick（华威大学）
Exponential improvements in Rado's covering problem
Istituto Nazionale di Alta Matematica “F. Severi”（高等数学国家研究所“F. Severi”）；Scuola Normale Superiore（比萨高等师范学院）；Algoresearch L.L.C.（Algoreresearch有限责任公司）；Alfréd Rényi Institute of Mathematics（阿尔弗雷德·雷尼数学研究所）；Research Institute of the University of Bucharest（布加勒斯特大学研究院）
In this paper we adapt work of Z.-D. Liu to prove a ball covering property for non-branching $\mathsf{CD}$ spaces with nonnegative curvature outside a compact set. As a consequence we obtain uniform bounds on the number of ends of such spaces.
