Orthogonal shadows and index of Grassmann manifolds

In this paper we study the $\mathbb{Z}\mathrm{/}2$ action on real Grassmann manifolds $G_n({\mathbb{R}}^{2n})$ and ${\tilde{G}}_n({\mathbb{R}}^{2n})$ given by taking (appropriately oriented) orthogonal complement. We completely evaluate the related $\mathbb{Z}\mathrm{/}2$ 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 $n=2^a(2b+1)$, $k=2^{a+1}-1$, $C$ a convex body in ${\mathbb{R}}^{2n}$, and $k$ real valued functions ${\alpha }_1,\dots ,{\alpha }_k$ 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 $1\le i\le k$.