
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 be a set of points on the plane in general position. We say that a set of convex polygons with vertices in is a convex decomposition of if: Union of all elements in is the convex hull of every element in is empty, and for any two different elements of their interiors are disjoint. A minimal convex decomposition of is a convex decomposition such that for any two adjacent elements in its union is a non convex polygon. It is known that always has a minimal convex decomposition with at most elements. Here we prove that always has a minimal convex decomposition with at most elements.
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Taxonomy
TopicsComputational Geometry and Mesh Generation · Point processes and geometric inequalities · graph theory and CDMA systems
