论文标题
对随机超平面镶嵌的尖锐估计
Sharp estimates on random hyperplane tessellations
论文作者
论文摘要
我们研究了在$ \ mathbb {r}^n $中产生任意集合$ t $的超平面镶嵌的问题，以确保任意两个点之间的欧几里得距离对应于将它们分开的超级分数，将它们分开至预先指定的错误$δ$。我们专注于随机分布偏移的随机高斯镶嵌，并在需要的超平面$ m $的数量上得出了急剧的界限。令人惊讶的是，我们的较低估计伪造了$ m \ sim \ ell _*^2（t）/δ^2 $的猜想，其中$ \ ell _*^2（t）$是$ t $的高斯宽度，是最佳的。
We study the problem of generating a hyperplane tessellation of an arbitrary set $T$ in $\mathbb{R}^n$, ensuring that the Euclidean distance between any two points corresponds to the fraction of hyperplanes separating them up to a pre-specified error $δ$. We focus on random gaussian tessellations with uniformly distributed shifts and derive sharp bounds on the number of hyperplanes $m$ that are required. Surprisingly, our lower estimates falsify the conjecture that $m\sim \ell_*^2(T)/δ^2$, where $\ell_*^2(T)$ is the gaussian width of $T$, is optimal.
