Bisecting masses with families of parallel hyperplanes

Abstract

We prove a common generalization of several mass partition results using hyperplane arrangements to split $\mathbb{R}^d$ into two sets. Our main result implies the ham sandwich theorem, the necklace splitting theorem for two thieves, a theorem about chessboard splittings using hyperplanes with fixed directions, and all known cases of Langerman's conjecture about bisections with $n$ hyperplanes. Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Soberón: \emph{For any $d+k-1$ measures in $\mathbb{R}^d$, there exist an arrangement of $k$ parallel hyperplanes that bisects each of the measures.} The general result follows from the case of measures that are supported on a finite set with an odd number of points. The proof for this case is inspired by ideas of differential and algebraic topology, but it is a completely elementary parity argument. Additionally, we disprove a conjecture by Langerman on bisections of measures using hyperplane arrangements, showing that the conditions in our main result are sometimes necessary.