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

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& Soberón, 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 \(d\geq j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}\).