References & Citations
Mathematics > Combinatorics
Title: Bisecting masses with families of parallel hyperplanes
(Submitted on 22 Apr 2024 (v1), last revised 29 Apr 2024 (this version, v2))
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.
Submission history
From: Pablo Soberón [view email][v1] Mon, 22 Apr 2024 16:31:32 GMT (107kb,D)
[v2] Mon, 29 Apr 2024 16:24:28 GMT (94kb,D)
Link back to: arXiv, form interface, contact.