We study the Tukey layers and convex layers of a planar point set, which consists of $n$ points independently and uniformly sampled from a convex polygon with $k$ vertices. We show that the expected number of vertices on the first $t$ Tukey layers is $O\left(kt\log(n/k)\right)$ and the expected number of vertices on the first $t$ convex layers is $O\left(kt^{3}\log(n/(kt^2))\right)$. We also show a lower bound of $\Omega(t\log n)$ for both quantities in the special cases where $k=3,4$. The implications of those results in the average-case analysis of two computational geometry algorithms are then discussed.

中文翻译：

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随机 Tukey 层和凸层的预期大小

我们研究了平面点集的 Tukey 层和凸层，该点集由从具有 $k$ 个顶点的凸多边形独立且均匀采样的 $n$ 个点组成。我们表明，第一个 $t$ Tukey 层上的预期顶点数为 $O\left(kt\log(n/k)\right)$ 并且前 $t$ 个凸层上的预期顶点数为 $O\left(kt\log(n/k)\right)$ O\left(kt^{3}\log(n/(kt^2))\right)$。在 $k=3,4$ 的特殊情况下，我们还显示了两个数量的 $\Omega(t\log n)$ 下限。然后讨论了这些结果在两种计算几何算法的平均情况分析中的含义。