  In this paper we study the $\Z/2$ action on real Grassmann manifolds
$G_{n}(\R^{2n})$ and $\widetilde{G}_{n}(\R^{2n})$ given by taking
(appropriately oriented) orthogonal complement. We completely evaluate the
related $\Z/2$ 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 $\R^{2n}$, and $k$ real valued functions
$\alpha_1,\ldots,\alpha_k$ continuous on convex bodies in $\R^{2n}$ with
respect to the Hausdorff metric, there exists a subspace $V\subseteq\R^{2n}$
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 $1\leq i\leq k$.
