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. (1998) [6].

我们研究了与球面超平面细分相关的几何特性 ${\mathbb{电阻}}^d$. 我们首先考虑欧几里得球体上的固定x和曲面细分$米\gg d$通过原点的超平面具有根据高斯分布分布的法向量。我们表明，很有可能存在一个基数为 的超平面子集$d\mathrm{日志}(d)\mathrm{日志}(米)$使得包含由这些超平面引起的x的单元格的半径由以上限制，最多为常数，$d\mathrm{日志}(d)\mathrm{日志}(米)/米$. 我们将此结果扩展为以高概率适用于曲面细分中的所有单元格。对于对数项，该上限与 Goyal 等人先前建立的下限相匹配。(1998) [6]。