Orthogonal shadows and index of Grassmann manifolds

Abstract

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 $α_1,\ldots,α_k$ 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 $α_i$, which is $α_i (p_V(C))=α_i (p_{V^\perp} (C))$ for all $1\leq i\leq k$.