A Mass Partition Problem Related to Equivariant Sections of Stiefel Bundles

We consider a geometric combinatorial problem naturally associated to the geometric topology of certain spherical space forms. Given a collection of m mass distributions on R^n, the existence of k linearly independent regular q-fans, each of which equipartitions each of the measures, can in many cases be deduced from the existence of a Z_q-equivariant section of the Stiefel bundle V_k(F^n) over S(F^n), where V_k(F^n) is the Stiefel manifold of orthonormal k-frames in F^n, F = R or C, and S(F^n) is the corresponding unit sphere. For example, the parallelizability of RP^{n-1}, n = 2,4,8, implies that any two measures on R^n can be bisected by (n - 1) pairwise-orthogonal hyperplanes, while the triviality of the S^1- bundle V_2(C^2)/Z_q over the standard Lens Spaces L^3(q), q = 3 or 4, yields that each measure on R^4 can be equipartitioned into regular q-sectors by a pair of orthogonal regular q-fans.