A general algorithm for convex fair partitions of convex polygons

• Published in: Fixed point theory and algorithms for sciences and engineering Vol. 2024; no. 1; pp. 13 - 19
• Main Authors: Campillo, Mathilda, Gonzalez-Lima, Maria D., Uribe, Bernardo
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 12.08.2024; Springer Nature B.V; SpringerOpen
• Subjects: Algorithms; Analysis; Applications of Mathematics; Centroidal Voronoi partition; Convex equipartition; Convexity; Differential Geometry; Fair partition; Lloyd’s algorithm; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Optimization and Real World Applications; Polygons; Topology; Voronoi partition; Convex equipartition; Fair partition; Voronoi partition; 52A38; 68U05; Lloyd’s algorithm; Centroidal Voronoi partition
• ISSN: 2730-5422; 2730-5422
• DOI: 10.1186/s13663-024-00769-y

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton’s method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd’s algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists.