TOPOLOGY OF THE GRÜNBAUM–HADWIGER–RAMOS PROBLEM FOR MASS ASSIGNMENTS

Abstract

In this paper, motivated by recent work of Schnider and Axelrod-Freed and Soberón, we study an extension of the classical Grünbaum–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 µ1, …, µj on the Grassmann manifold GℓRd) and (a given integer k ≥ 1 there exist a linear subspace L ∈ GℓRd) and k affine hyperplanes in L that equipart the masses µL1, …, µLj assigned to the subspace L, provided that d ≥ j + (2k−1 − 1)2⌊log 2j⌋.