Papyrus 41

Template:Distinguish2 In mathematics, the $\varepsilon$-covering number of a set $K$, denoted by $C(K,\varepsilon )$, in a metric space $(X,d)$ for some $\varepsilon >0$ is the minimum number of balls of radius $\varepsilon$ that are needed to cover $K$. For example, when $(X,d)$ is the $n$-dimensional Euclidean space and $K$ is the unit Euclidean ball, $C(K,\varepsilon )=O((2/\varepsilon )^{n})$.

A related concept is the $\varepsilon$-packing number which is defined as the maximum number of disjoint balls of radius $\varepsilon /2$ that fit into $K$.

The covering number $C(K,\varepsilon )$ and the packing number $P(X,\varepsilon )$ are related by the following inequalities:

$C(K,2\varepsilon )\leq P(K,2\varepsilon )\leq C(K,\varepsilon )$ .

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