I'm trying to analyze the run time of an algorithm that returns a set of convex hulls, in layers, on a set of points in $\mathbb{R}^2$. I have a subroutine of finding the convex hull, which runs on the order of $O(n\log h)$. $n$ represents the number of input points and $h$ is the number of edges on the convex hull. The idea is that you compute the convex hull, and then recursively call it on the interior points. How should I think about the run time when my points are being reduced by some $3 \leq h_i \leq n$ at every recursive step? I feel it looks something like this:
$n\log h_0 + (n-h_0)\log h_1 + (n-h_0-h_1)\log h_2 + \cdots + (n - \sum_{i=0}^{m-1}h_i)\log h_m + c$
But I'm unsure how to view it from big O notation or how to get it started.