Index of Grassmann manifolds and orthogonal shadows

Abstract

In this paper, we study the Z/2 action on the real Grassmann manifolds Gn (R2n) and ∼Gn (R2n) given by taking the (appropriately oriented) orthogonal complement.We completely evaluate the relatedZ/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 = 2a (2b + 1), k = 2a+1-1, a convex body C in R 2n, and k real-valued functions α1, . . . , αk continuous on convex bodies in R2n with respect to the Hausdorff metric, there exists a subspace V ⊆ R 2n such that projections of C to V and its orthogonal complement V have the same value with respect to each function αi, that is, αi(pV(C)) = αi(pV (C)) for all 1 ≤ i ≤ k.