  We study a geometric property related to spherical hyperplane tessellations
in $\mathbb{R}^{d}$. We first consider a fixed $x$ on the Euclidean sphere and
tessellations with $M \gg d$ hyperplanes passing through the origin having
normal vectors distributed according to a Gaussian distribution. We show that
with high probability there exists a subset of the hyperplanes whose
cardinality is on the order of $d\log(d)\log(M)$ such that the radius of the
cell containing $x$ induced by these hyperplanes is bounded above by, up to
constants, $d\log(d)\log(M)/M$. We extend this result to hold for all cells in
the tessellation with high probability. Up to logarithmic terms, this upper
bound matches the previously established lower bound of Goyal et al. (IEEE T.
Inform. Theory 44(1):16-31, 1998).
