Expected size of random Tukey layers and convex layers

Abstract

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(ktlog⁡(n/k)) and the expected number of vertices on the first t convex layers is O(kt3log⁡(n/(kt2))). We also show a lower bound of Ω(tlog⁡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.