Convex fair partitions into an arbitrary number of pieces

TL;DR

This paper proves that any convex shape in the plane can be divided into an arbitrary number of convex parts with equal areas and perimeters, extending previous results limited to prime powers.

Contribution

It generalizes existing partition results to any integer number of parts and discusses potential higher-dimensional extensions and challenges.

Findings

Any convex planar body can be partitioned into m convex parts with equal areas and perimeters for all m ≥ 2.

The result extends previous work limited to prime power divisions.

Discussion of higher-dimensional generalizations and limitations of current techniques.

Abstract

We prove that any convex body in the plane can be partitioned into $m$ convex parts of equal areas and perimeters for any integer $m\ge 2$; this result was previously known for prime powers $m=p^k$. We also discuss possible higher-dimensional generalizations and difficulties of extending our technique to equalizing more than one non-additive function.