 Unit sphere net from ball covering

Suppose the unit ball $B$ of a $k$-dimensional normed vector space has metric covering number $N(r,B)\le (A/r{)}^k$ at $r=\delta /2$, where $A,\delta >0$. Cover $B$ by at most $(2A/\delta {)}^k$ balls of radius $\delta /2$. Discard every ball missing the unit sphere, and select one point of the unit sphere from each remaining ball. These selected points form a radius-$\delta$ metric net $N_{\delta }$ of the unit sphere, withIndeed, a unit sphere point and the selected point from the same covering ball each lie within $\delta /2$ of its center, so the triangle inequality bounds their distance by $\delta$. The selection ensures that all centers of the new metric net have unit norm; the original covering centers need not. For Euclidean balls, the volumetric bound for Euclidean metric nets supplies the initial metric covering number estimate.