Orthogonal shadows and index of Grassmann manifolds

In this paper we study the $\mathbb Z/2$ action on real Grassmann manifolds $G_{n}(\mathbb R^{2n})$ and $\widetilde{G}_{n}(\mathbb R^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\mathbb Z/2$ Fadell--Husseini index utilizing a novel computation of the Stiefel--Whitney classes of the wreath product of a vector bundle. These results are used to establish the following geometric result about the orthogonal shadows of a convex body: For $n=2^a (2 b+1)$, $k=2^{a+1}-1$, $C$ a convex body in $\mathbb R^{2n}$, and $k$ real valued functions $\alpha_1,\ldots,\alpha_k$ continuous on convex bodies in $\mathbb R^{2n}$ with respect to the Hausdorff metric, there exists a subspace $V\subseteq\mathbb R^{2n}$ such that projections of $C$ to $V$ and its orthogonal complement $V^{\perp}$ have the same value with respect to each function $\alpha_i$, which is $\alpha_i (p_V(C))=\alpha_i (p_{V^\perp} (C))$ for all $1\leq i\leq k$.