Volumetric bound for Euclidean metric nets
= Volumetric bound for Euclidean metric nets
{title2=$|U|\leq(1+2R/\varepsilon)^n$}
A subset of the radius-$R$ <Euclidean ball> admits an $\varepsilon$-<metric net> of size at most $(1+2R/\varepsilon)^n$. Take a maximal separated set: its disjoint radius-$\varepsilon/2$ <Euclidean balls> lie in the radius-$(R+\varepsilon/2)$ ball. Comparing <Lebesgue measures> gives the estimate. In particular, the <unit sphere> admits a net of radius one with at most $3^n$ points.