Bisecting masses with families of parallel hyperplanes

TL;DR

This paper generalizes several mass partition theorems using hyperplane arrangements, proving new cases and confirming conjectures through elementary parity arguments inspired by topology.

Contribution

It introduces a unified framework for mass partition results, extending known theorems and confirming new cases of a conjecture on bisecting measures with parallel hyperplanes.

Findings

Generalizes ham-sandwich and necklace splitting theorems

Proves new cases of Langerman's conjecture

Confirms infinite cases of Takahashi and Soberón's conjecture

Abstract

We prove a common generalization to 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 with hyperplanes with fixed directions, and all known cases of Langerman's conjecture about equipartitions with $n$ hyperplanes. Our main result also confirms an infinite number of previously unknown cases of the following conjecture of Takahashi and Sober\'on: 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.