Topology of the Grünbaum--Hadwiger--Ramos problem for mass assignments

2022/08/01 by Pavle V. M. Blagojević, Blagojević, Pavle V. M., Jaime Calles Loperena +5 · 1 citation
Mathematics · #Advanced Topology and Set Theory #Algebraic Topology (math.AT) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2208.00666

openalex publication_date 2022/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, motivated by recent work of Schnider and Axelrod-Freed & Soberón, we study an extension of the classical Grünbaum--Hadwiger--Ramos mass partition problem to mass assignments. Using the Fadell--Husseini index theory we prove that for a given family of j mass assignments μ₁,…,μ_j on the Grassmann manifold G_ℓ(\R^d) and a given integer k≥ 1 there exist a linear subspace L∈ G_ℓ(\R^d) and k affine hyperplanes in L that equipart the masses μ₁^L,…,μ_j^L assigned to the subspace L, provided that d≥ j + (2^k-1-1)2^{\lfloorlog₂j\rfloor}.

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