Index of Grassmann manifolds and orthogonal shadows

<jats:p>In this paper, we study the <jats:inline-formula id="j_forum-2018-0058_ineq_9999"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0569. png" /> <jats:tex-math>{\mathbb{Z}/2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> action on the real Grassmann manifolds <jats:inline-formula id="j_forum-2018-0058_ineq_9998"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0445. png" /> <jats:tex-math>{G_{n}(\mathbb{R}^{2n})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula id="j_forum-2018-0058_ineq_9997"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0643. png" /> <jats:tex-math>{\widetilde{G}_{n}(\mathbb{R}^{2n})}</jats:tex-math> </jats:alternatives> </jats:inline-formula> given by taking the (appropriately oriented) orthogonal complement. We completely evaluate the related <jats:inline-formula id="j_forum-2018-0058_ineq_9996"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0569. png" /> <jats:tex-math>{\mathbb{Z}/2}</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:inline-formula id="j_forum-2018-0058_ineq_9995"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0757. png" /> <jats:tex-math>{n=2^{a}(2b+1)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula id="j_forum-2018-0058_ineq_9994"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0734. png" /> <jats:tex-math>{k=2^{a+1}-1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, a convex body <jats:italic>C</jats:italic> in <jats:inline-formula id="j_forum-2018-0058_ineq_9993"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0564. png" /> <jats:tex-math>{\mathbb{R}^{2n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and <jats:italic>k</jats:italic> real-valued functions <jats:inline-formula id="j_forum-2018-0058_ineq_9992"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0526. png" /> <jats:tex-math>{\alpha_{1},\ldots,\alpha_{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> continuous on convex bodies in <jats:inline-formula id="j_forum-2018-0058_ineq_9991"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0564. png" /> <jats:tex-math>{\mathbb{R}^{2n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with respect to the Hausdorff metric, there exists a subspace <jats:inline-formula id="j_forum-2018-0058_ineq_9990"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0514. png" /> <jats:tex-math>{V\subseteq\mathbb{R}^{2n}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> such that projections of <jats:italic>C</jats:italic> to <jats:italic>V</jats:italic> and its orthogonal complement <jats:inline-formula id="j_forum-2018-0058_ineq_9989"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0516. png" /> <jats:tex-math>{V^{\perp}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> have the same value with respect to each function <jats:inline-formula id="j_forum-2018-0058_ineq_9988"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0529. png" /> <jats:tex-math>{\alpha_{i}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, that is, <jats:inline-formula id="j_forum-2018-0058_ineq_9987"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0528. png" /> <jats:tex-math>{\alpha_{i}(p_{V}(C))=\alpha_{i}(p_{V^{\perp}}(C))}</jats:tex-math> </jats:alternatives> </jats:inline-formula> for all <jats:inline-formula id="j_forum-2018-0058_ineq_9986"> <jats:alternatives> <jats:inline-graphic xlink:href="graphic/j_forum-2018-0058_eq_0369. png" /> <jats:tex-math>{1\leq i\leq k}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. </jats:p>