Topology of the Gr\"unbaum--Hadwiger--Ramos problem for mass assignments

TL;DR

This paper extends the classical mass partition problem to mass assignments on Grassmann manifolds, proving the existence of hyperplanes that equipartition multiple masses under certain dimensional constraints using Fadell--Husseini index theory.

Contribution

It introduces a new topological approach to the mass assignment problem, generalizing previous results and providing explicit conditions for equipartition in higher dimensions.

Findings

Existence of hyperplanes that equipartition multiple mass assignments.

Application of Fadell--Husseini index theory to mass partition problems.

Derived explicit bounds on the dimension for the existence of equipartitions.

Abstract

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& Sober\'on, we study an extension of the classical Gr\"unbaum--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 $\mu_1,\dots,\mu_j$ on the Grassmann manifold $G_{\ell}(\R^d)$ and a given integer $k\geq 1$ there exist a linear subspace $L\in G_{\ell}(\R^d)$ and $k$ affine hyperplanes in $L$ that equipart the masses $\mu_1^L,\dots,\mu_j^L$ assigned to the subspace $L$, provided that $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.