Orthogonal shadows and index of Grassmann manifolds

TL;DR

This paper investigates the $ ext{Z}/2$ action on real Grassmann manifolds, computes the associated Fadell–Husseini index using Stiefel–Whitney classes, and applies these results to a geometric problem involving orthogonal shadows of convex bodies.

Contribution

It provides a complete evaluation of the $ ext{Z}/2$ Fadell–Husseini index for certain Grassmann manifolds using a novel Stiefel–Whitney class computation, and applies this to a geometric shadow problem.

Findings

Established the $ ext{Z}/2$ Fadell–Husseini index for specific Grassmann manifolds.

Proved the existence of a subspace with equal projections for convex bodies and functions.

Connected topological index computations to geometric shadow properties.

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 $\alpha_1,\ldots,\alpha_k$ 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 $1\leq i\leq k$.