Bisections of Mass Assignments Using Flags of Affine Spaces
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Author(s)
Axelrod-Freed, Ilani
Soberón, Pablo
Date Issued
December 30, 2022
Journal
Discrete & Computational Geometry
Publisher
Springer US
Citation
Axelrod-Freed, I., Soberón, P. Bisections of Mass Assignments Using Flags of Affine Spaces. Discrete Comput Geom 72, 550–568 (2024).
Version
Author's final manuscript
Abstract
We use recent extensions of the Borsuk–Ulam theorem for Stiefel manifolds to generalize the ham sandwich theorem to mass assignments. A k-dimensional mass assignment continuously imposes a measure on each k-dimensional affine subspace of ℝ𝑑
. Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces 𝑆𝑘−1⊂𝑆𝑘⊂…⊂𝑆𝑑−1⊂ℝ𝑑
such that 𝑆𝑖
bisects all the mass assignments on 𝑆𝑖+1
for every i. We show it is possible to do so whenever the number of mass assignments of dimensions (𝑘,…,𝑑)
is a permutation of (𝑘,…,𝑑)
. We extend previous work on mass assignments and the central transversal theorem. We also study the problem of halving several families of (𝑑−𝑘)
-dimensional affine spaces of ℝ𝑑
using a (𝑘−1)
-dimensional affine subspace contained in some translate of a fixed k-dimensional affine space. For 𝑘=𝑑−1
, there results can be interpreted as dynamic ham sandwich theorems for families of moving points.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00454-022-00465-x
