  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.
