Simultaneous packing and covering in sequence spaces

TL;DR

This paper adapts Klee's construction to demonstrate an optimal packing of unit balls in sequence spaces, where enlarging the balls beyond a certain radius ensures complete coverage of the space.

Contribution

It introduces a novel packing construction in $ ext{ell}_p$ spaces and proves the optimality of the covering radius for enlarged balls.

Findings

Constructed an efficient packing of unit balls in $ ext{ell}_p$ spaces.

Proved the minimal radius needed for enlarged balls to cover the entire space.

Established the optimality of the radius $2^{1-1/p}$ for covering.

Abstract

We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) 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.