Partitions of mass assignments with spheres and wedges

2025/07/09 by Deron Lessure, Pablo Soberón, Lessure, Deron +1
Computer Science · Mathematics · #28A75 #52A37 #52C35 #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2507.06919

openalex publication_date 2025/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we generalize classic mass partition results dealing with partitions using spheres, parallel hyperplanes, or axis-parallel wedges to the setting of mass assignments. In a mass assignment problem, we assign mass distributions continuously to all k-dimensional subspaces of ℝ^d, and seek to guarantee the existence of a particular subspace in which more masses can be bisected than those by analyzing the problem in ℝ^k. We prove new mass assignment results for spheres, parallel hyperplanes, and axis-parallel wedges. The proof techniques rely on new Borsuk--Ulam type theorems on spheres and Stiefel manifolds.

Citations

• Bisections of mass assignments by parallel hyperplanes
• Bisecting masses with families of parallel hyperplanes
• A Center Transversal Theorem for mass assignments
• Many partitions of mass assignments
• Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments
• Bisections of mass assignments using flags of affine spaces
• Lifting methods in mass partition problems
• A survey of mass partitions
• Equipartitions and Mahler volumes of symmetric convex bodies
• Ham-Sandwich cuts and center transversals in subspaces
• Borsuk-Ulam theorems for products of spheres and Stiefel manifolds\n revisited
• Measure Partitions Using Hyperplanes with Fixed Directions

Related

• Many partitions of mass assignments
• A survey of mass partitions
• Bisecting masses with families of parallel hyperplanes
• Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments
• Bisections of mass assignments using flags of affine spaces