  In this paper we compute: the Schwarz genus of the Stiefel manifold
$V_k(\mathbb R^n)$ with respect to the action of the Weyl group $W_k:=(\mathbb
Z/2)^{k}\rtimes\Sigma_k$, and the Lusternik--Schnirelmann category of the
quotient space $V_k(\mathbb R^n)/W_k$. Furthermore, these results are used in
estimating the number of: critically outscribed parallelotopes around the
strictly convex body, and Birkhoff--James orthogonal bases of the normed finite
dimensional vector space.
