Simultaneous Packing and Covering in Sequence Spaces

Abstract

ABSTRACT. We adapt a construction of Klee (1981) to find a packing of unit balls in ℓp (1 ≤ p < ∞) which is efficient in the sense that enlarging the radius of each ball to any R > 2 1−1/p covers the whole space. We show that the value 2 1−1/p is optimal.

Density bounds for outer parallel domains of unit ball packings

“…(We note that this notion is a Euclidean analogue of the notion of perfectly distributed points on a sphere introduced by L. Fejes Tóth in [15], which is different from the notion of universally optimal distribution of points on spheres introduced by Cohn and Kumar in [12].) Recall that µ d > 0 is called the simultaneous packing and covering constant of the closed unit ball B d = {x ∈ E d | x ≤ 1} in E d if the following holds (for more details see for example, [27]): µ d > 0 is the smallest positive real number µ such that there is a unit ball packing P := {c i + B d | i = 1, 2, . .…”

Density Bounds for Outer Parallel Domains of Unit Ball Packings

“…if the following holds (for more details see for example, [27]): µ d > 0 is the smallest positive real number µ such that there is a unit ball packing P :…”