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

Abstract

In this paper, motivated by recent work of Schnider and Axelrod-Freed \& 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 ${\mu }_1,\dots ,{\mu }_j$ on the Grassmann manifold $G_{\ell }({\mathbb{R}}^d)$ and a given integer $k\ge 1$ there exist a linear subspace $L\in G_{\ell }({\mathbb{R}}^d)$ and $k$ affine hyperplanes in $L$ that equipart the masses ${\mu }_1^L,\dots ,{\mu }_j^L$ assigned to the subspace $L$, provided that $d\ge j+(2^{k-1}-1)2^{\lfloor {\log }_{2}j\rfloor }$.