Topology of the grunbaum-hadwiger-ramos problem for mass assignments

In this paper, motivated by recent work of Schnider and Axelrod-Freed and Soberon, we study an extension of the classical Grunbaum-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), ... , mu(j) on the Grassmann manifold G(l) (R-d) and a given integer k >= 1 there exist a linear subspace L is an element of G(l) (R-d) and k affine hyperplanes in L that equipart the masses mu(L)(1), ... , mu(L)(j) assigned to the subspace L, provided that d >= j + (2(k-1) - 1)(left perpendicular Llog2 j right perpendicular).