The Schwarz genus of the Stiefel manifold

In this paper we compute: the Schwarz genus of the Stiefel manifold Vk(Rn)Vk(Rn) with respect to the action of the Weyl group Wk:=(Z/2)k⋊SkWk:=(Z/2)k⋊Sk, and the Lusternik–Schnirelmann category of the quotient space Vk(Rn)/WkVk(Rn)/Wk. 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.