Cookie cutters: Bisections with fixed shapes

TL;DR

This paper investigates equitable mass partitions using scaled copies of a shape in Euclidean space, extending known results to star-shaped sets and specific instances like hypercubes, with new topological methods.

Contribution

It introduces new topological Borsuk--Ulam-type theorems to establish simultaneous bisections for multiple measures with general shapes.

Findings

Positive results for bisection of any d+1 measures with star-shaped sets

Extension of results to hypercubes and cylinders

Answers to conjecture by Soberón and Takahashi

Abstract

In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $\mathbb{R}^d$. In this manuscript, we study the problem of finding simultaneous bisections of measures using scaled copies of a prescribed set $K$. We distinguish the problem when we are allowed to use scaled and translated copies of $K$ and the problem when we are allowed to use scaled isometric copies of $K$. These problems have only previously been studied if $K$ is a half-space or a Euclidean ball. We obtain positive results for simultaneous bisection of any $d+1$ masses for star-shaped compact sets $K$ with non-empty interior, where the conditions on the problem depend on the smoothness of the boundary of $K$. Additional proofs are included for particular instances of $K$, such as hypercubes and cylinders, answering positively a conjecture of Sober\'on and Takahashi. The proof methods are topological and involve new Borsuk--Ulam-type theorems.