2024/12/05 by Nikola Sadovek, Pablo Soberón, Sadovek, Nikola +1 · 1 citation
Mathematics · #52A37 #52C35 #55N91 #55R91 #Advanced Mathematical Identities #Advanced Mathematical Theories #Advanced Statistical Methods and Models #Algebraic Topology (math.AT) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2412.04058
openalex publication_date 2024/12/05 · openalex created_date 2024/12/07 · openalex updated_date 2026/07/28
In this paper, we prove a result on the bisection of mass assignments by parallel hyperplanes on Euclidean vector bundles. Our methods consist of the development of a novel lifting method to define the configuration space--test map scheme, which transforms the problem to a Borsuk--Ulam-type question on equivariant fiber bundles, along with a new computation of the parametrized Fadell--Husseini index. As the primary application, we show that any d+k+m-1 mass assignments to linear d-spaces in ℝd+m can be bisected by k parallel hyperplanes in at least one d-space, provided that the Stirling number of the second kind S(d+k+m-1, k) is odd. This generalizes all known cases of a conjecture by Soberón and Takahashi, which asserts that any d+k-1 measures in ℝd can be bisected by k parallel hyperplanes.