Peeling sequences: a directional method for the three-block construction

Abstract

A peeling sequence of a finite planar point set is an ordering of its points, such that when the points get removed in this order, each removed point is a vertex of the current convex hull of the set. Write $g(S)$ for the number of such sequences, and $g(n)$ for the minimum of $g(S)$ over $n$-point sets in general position. We present a method which can be used to prove better upper bounds on the previously analysed recursive 3-block constructions $S_n$. In fact, we prove $g(n)\le g(S_n)=O(6.57^n)$, using directional restrictions and a weighted prefix-tree form of the Kraft--McMillan inequality. From the other direction, we show that the same construction cannot attain the current best known lower bound of $c\cdot 3^n$. In fact, $g(S_n)\ge 5.22^n$ when $n$ is a large enough power of $3$. Improving the general lower bound on $g(n)$ is still an open question.

arXiv comment18 pages, 7 figures