Bisecting masses with families of parallel hyperplanes
arxiv(2024)
摘要
We prove a common generalization to several mass partition results using
hyperplane arrangements to split 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 Soberón:
For any d+k-1 measures in 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.
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