Bisections of mass assignments using flags of affine spaces

2021/09/27 by Ilani Axelrod-Freed, Pablo Soberón, Axelrod-Freed, Ilani +1 · 1 citation
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2109.13106

openalex publication_date 2021/09/27 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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 ℝ^d. Given a finite collection of mass assignments of different dimensions, one may ask if there is some sequence of affine subspaces S_k-1 ⊂ S_k ⊂ … ⊂ S_d-1 ⊂ ℝ^d such that S_i bisects all the mass assignments on S_i+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 ℝ^d 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.

Cited by

• Partitions of mass assignments with spheres and wedges

Related

• Ham-Sandwich cuts and center transversals in subspaces
• Partitions of mass assignments with spheres and wedges
• Many partitions of mass assignments
• Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments
• A Center Transversal Theorem for mass assignments