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

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 $d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.