Further analysis of Peeling Sequences

TL;DR

This work studies peeling sequences in planar point sets in general position by formalizing $g(P)$ and $g(n)=\min_{|P|=n} g(P)$ and improving the known upper bound on the minimal number of peeling sequences from $O(12.29^n)$ to $O(9.78^n)$. The authors introduce two recursive constructions, $S_n$ and $B_n$, to encode worst-case peeling behavior, and develop a decomposition framework together with entropy-based and binomial bounds to perform a detailed case analysis. By inductively bounding the number of peeling sequences across carefully partitioned scenarios and showing the total contribution of all cases stays below a constant $1/500$, they establish $g(n) \le \frac{9.78^n}{500}$ for $n \ge 6$. The approach combines the $S_n$/$B_n$ constructions with a suite of lemmas on simplified peeling sequences and combinatorial bounds, refining the analysis of convex-layer peeling and providing a near-tight exponential bound; the paper also highlights open questions on strengthening lower bounds.

Abstract

Let $P\subset \mathbf{R}^2$ be a set of $n$ points in general position. A peeling sequence of $P$ is a list of its points, such that if we remove the points from $P$ in that order, we always remove the next point from the convex hull of the remainder of $P$. Using the methodology of Dumitrescu and Tóth \cite{Dumitrescu}, with more careful analysis, we improve the upper bound on the minimum number of peeling sequences for an $n$ point set in the plane from $\frac{12.29^n}{100}$ to $\frac{9.78^n}{500}$.

Figures (4)

• Figure 1: $S'_n$ before the flattening step
• Figure 2: $S_n$ after the flattening step
• Figure 3: The construction of $B_n$. On the ray of the x-axis, we remove $2$ subrays of the corresponding recursive set from $S_n$.
• Figure 4: An unflattened representation of $B_{18}$, with a labeling of the starting elements of a peeling sequence. For simplicity, we put vertices on the same ray as collinear, even though they not actually are. Here, the first ray disappears after the $7$th removed element, hence $s=7$ would hold for this sequence.

Theorems & Definitions (13)

• Definition
• Theorem 1
• Definition
• Lemma 2
• Lemma 3
• Lemma 4
• proof
• Definition
• Lemma 5
• proof