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.

翻译：我们研究一个平面点组的图基层和锥形层,它由独立和统一地从一个带有$k$的锥形多边形上独立抽样的零点组成。我们显示,第一批图基层的预期脊椎数是$O\left(kt\log(n/k)\right)$,第一批美元锥形层的预期脊椎数是$O\left(kt ⁇ 3}log(n/(kt ⁇ 2))\right)$。在特例中,如果是 $k=3,4美元,我们还显示两种数量的Omega(t\log n)$的下限值。然后讨论这些结果对两种计算几何算算算算法平均分析的影响。