The Schwarz genus of the Stiefel manifold and counting geometric configurations

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 $V_k({\mathbb{R}}^n)\mathrm{/}{W}_k$. 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.