Partitions of mass assignments with spheres and wedges

In this paper, we generalize classic mass partition results dealing with partitions using spheres, parallel hyperplanes, or axis-parallel wedges to the setting of mass assignments. In a mass assignment problem, we assign mass distributions continuously to all 𝑘 -dimensional subspaces of ℝ 𝑑 , and seek to guarantee the existence of a particular subspace in which more masses can be bisected than those by analyzing the problem in ℝ 𝑘 . We prove new mass assignment results for spheres, parallel hyperplanes, and axis-parallel wedges. The proof techniques rely on new Borsuk–Ulam type theorems on spheres and Stiefel manifolds.