Cookie cutters: Bisections with fixed shapes

Abstract

In a mass partition problem, we are interested in finding equitable partitions of smooth measures in $R^d$. 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 Soberón and Takahashi. The proof methods are topological and involve new Borsuk–Ulam-type theorems.