The Schwarz genus of the Stiefel manifold

In this paper we compute: the Schwarz genus of the Stiefel manifold V k ( R n ) with respect to the action of the Weyl group W k : = ( Z / 2 ) k ⋊ S k , and the Lusternik–Schnirelmann category of the quotient space V k ( R n ) / W k . Furthermore, these results are used in estimating the number of critically outscribed parallelotopes around a strictly convex body, and Birkhoff–James orthogonal bases of a normed finite dimensional vector space.