Convex decompositions of point sets in the plane

TL;DR

This paper proves that any set of points in the plane can be convexly decomposed into a limited number of polygons, with an upper bound related to the number of interior and boundary points.

Contribution

It establishes a new upper bound on the size of convex decompositions for point sets in the plane, improving understanding of geometric partitioning.

Findings

Existence of convex decompositions with at most (4/3)|I(P)| + (1/3)|B(P)| + 1 polygons.

Upper bound of (4/3)|P| - 2 polygons for convex decompositions.

Provides a constructive approach to achieve the bound.

Abstract

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$, and no point of $P$ lies in the interior of the polygons. We show that there is a convex decomposition of $P$ with at most $\frac{4}{3}|I(P)|+\frac{1}{3}|B(P)|+1\le \frac{4}{3}|P|-2$ elements, where $B(P)\subseteq P$ is the set of points at the vertices of the convex hull of $P$, and $I(P)=P-B(P)$.