Covering of high-dimensional sets

Abstract

Let $(\mathcal{X},ρ)$ be a metric space and $λ$ be a Borel measure on this space defined on the $σ$-algebra generated by open subsets of $\mathcal{X}$; this measure $λ$ defines volumes of Borel subsets of $\mathcal{X}$. The principal case is where $\mathcal{X} = \mathbb{R}^d$, $ρ$ is the Euclidean metric, and $λ$ is the Lebesgue measure. In this article, we are not going to pay much attention to the case of small dimensions $d$ as the problem of construction of good covering schemes for small $d$ can be attacked by the brute-force optimization algorithms. On the contrary, for medium or large dimensions (say, $d\geq 10$), there is little chance of getting anything sensible without understanding the main issues related to construction of efficient covering designs.