Bisecting masses with families of parallel hyperplanes

We prove a common generalization to several mass partition results using hyperplane arrangements to split Rᵈ 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 Rᵈ, 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.