Minimal Convex Decompositions

TL;DR

This paper proves that for any set of points in the plane, there exists a minimal convex decomposition with at most 10n/7 polygons, improving previous bounds and providing a tighter structural understanding.

Contribution

The paper establishes a new upper bound of 10n/7 on the size of minimal convex decompositions for point sets, refining prior known bounds.

Findings

Established a new upper bound of 10n/7 for minimal convex decompositions.

Proved that such decompositions always exist for any point set in general position.

Improved upon the previous bound of 3n/2 elements.

Abstract

Let $P$ be a set of $n$ points on the plane in general position. We say that a set $\Gamma$ of convex polygons with vertices in $P$ is a convex decomposition of $P$ if: Union of all elements in $\Gamma$ is the convex hull of $P,$ every element in $\Gamma$ is empty, and for any two different elements of $\Gamma$ their interiors are disjoint. A minimal convex decomposition of $P$ is a convex decomposition $\Gamma'$ such that for any two adjacent elements in $\Gamma'$ its union is a non convex polygon. It is known that $P$ always has a minimal convex decomposition with at most $\frac{3n}{2}$ elements. Here we prove that $P$ always has a minimal convex decomposition with at most $\frac{10n}{7}$ elements.