Topology of the Grünbaum-Hadwiger-Ramos problem for mass assignments - Blagojević, Pavle V. M.; Loperena, Jaime Calles; Crabb, Michael C.; Dimitrijević Blagojević, Aleksandra S. (Topological Methods in Nonlinear Analysis, 2023)
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$μ_1,…,μ_j$ on the Grassmann manifold $G_{\ell}\big(\mathbb{R}^d\big)$ and a giveninteger $k≥ 1$ there exist a linear subspace $L\in G_{\ell}\big(\mathbb{R}^d\big)$ and$k$affine hyperplanes in $L$ that equipart the masses $μ_1^L,…,μ_j^L$assigned to the subspace $L$, provided that $d≥ j + (2^{k-1}-1)2^{\lfloor\log_2j\rfloor}$.