Orthogonal shadows and index of Grassmann manifolds

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 \(1\leq i\leq k\).