Convex Regions and their 'Fairest' Equipartitioning Fans

TL;DR

This paper studies the problem of partitioning convex regions into equal-area convex pieces using fans, aiming to make the perimeters of these pieces as uniform as possible, and explores properties and open questions related to such 'fairest' equipartitions.

Contribution

It introduces the concept of 'fairest equipartitioning fans' for convex regions and analyzes their basic properties, raising new questions for further research.

Findings

Basic properties of fairest equipartitioning fans identified

Conditions for minimal perimeter variation among pieces discussed

Open questions on optimal fan configurations posed

Abstract

A k-fan is a set of k half-lines (rays) all starting from the same point, called the origin of the fan. We discuss the partition of convex 2D regions into n (a positive integer) equal area convex pieces by fans with the following additional requirement: the perimeters of the resultant equal area pieces should be as close to one another as possible. We present some basic properties of such fans, which we call 'fairest equipartitioning fans', and raise further questions.