Borsuk-Ulam type theorem for Stiefel manifolds and orthogonal mass partitions

A generalization of the Borsuk-Ulam theorem to Stiefel manifolds is considered. This theorem is applied to derive bounds on $d$ that guarantee-for a given set of $m$ measures in ${\mathbb{R}}^d$-the existence of $k$ mutually orthogonal hyperplanes, any $n$ of which partition each of the measures into $2^n$ equal parts. If $n=k$, the result corresponds to the bound obtained in [14], but with the stronger conclusion that the hyperplanes are mutually orthogonal.