Covering the unit sphere with unit balls Consider the unit sphere $\mathbb{S}^{d-1}\subset \mathbb{R}^d$. Show that there is a constant $N(d)$ such that whenever $\{y_1,\ldots y_k\}\subset\mathbb{S}^{d-1}$ such that $\|y_j-y_j\|_2\geq 1$, then  $k\leq N(d)$. This means that the the number of balls of radius one that cover $\mathbb{S}^{d-1}$ and whose  centers have angles of at least $\pi/3$ is bounded by $N(d)$.

This result is used in the course of proving the Besicovitch covering theorem and in many places in the literature, the segment above is mention as a matter of fact. I believe the result to be true but a proof escapes me at the moment.

For $y\in S^{n-1}$ let $B(y)=\{z\in S^{n-1}:\|z-y\|_2<1/2\}$. This is a "spherical cap" inside $S^{n-1}$ and has non-zero measure in the "surface area" of $S^{n-1}$.
This measure is independent of $y$.
The condition $\|y_i-y_j\|_2\ge1$ means that the $B(y_i)$ are disjoint. But the measure
of $S^{n-1}$ is finite. So the number of $y_i$ is bounded by the measure of $S^{n-1}$ divided by the measure of each $B(y)$.
