INDEX OF GRASSMANN MANIFOLDS AND ORTHOGONAL SHADOWS DJORDJE BARALIĆ, PAVLE V. M. BLAGOJEVIĆ, ROMAN KARASEV, AND ALEKSANDAR VUČIĆ Abstract. In this paper we study the Z/2 action on real Grassmann manifolds Gn(R 2n ) and  Gn(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 (2b + 1), k =2 a+1 - 1, C a convex body in R 2n , and k real valued functions α 1 ,...,α k continuous on convex bodies in R 2n 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 α i , which is α i (p V (C)) = α i (p V ⊥(C)) for all 1 ≤ i ≤ k. 1. Introduction The Grassmann manifold of all n dimensional linear subspaces in the vector space V over some field is one of the classical and widely studied objects of algebraic topology with important applications in differential geometry and algebraic geometry. In this paper we study a particular free Z/2 action on a real Grassmann manifolds induced by taking orthogonal complements and use its properties to present an interesting geometric application. The main tool in the study of this action is the ideal valued index theory of Fadell and Husseini. A variety of different index theories, for the study of the non-existence of equivariant maps, were introduced and considered over the last seven decades. In 1952 Krasnosel’ski ˘ i[20] introduced a numerical Z/2-index for the subsets of spheres, which he called genus. He used genus to estimate the number of critical points of a particular class of weakly continuous functionals on a spheres in a Hilbert space; consult also [21]. Yang in 1955 introduced yet another Z/2-index calling it B-index with the aim of obtaining results of Borsuk–Ulam type [33]. The work of Conner and Floyd followed at the beginning of ’60s and the co-index was introduced [8, 9]. At the end of ’80s Fadell and Husseini in series of papers introduced a general index theory and applied it on various problems [11, 12, 13, 14, 15]. Further refinements of the Fadell and Husseini work were done by Volovikov in [32]. Since then computation of a concrete index for the given space and the given action was, and still is, a formidable, and by now a classical, problem. For example, the work of Volovikov [31], Petrović [28], Hara [16], Crabb [10], Blagojević and Karasev [2], Petrović and Prvulović [29], Blagojević, Lück and Ziegler [3], Simon [30] illustrate wide diversity of novel ideas and methods needed to be employed in course of computation of different Fadell–Husseini indexes. Let n ≥ 1 and 1 ≤ k ≤ n be integers. The real Grassmann manifold of all n dimensional linear subspaces in the Euclidean space R n+k is denoted by G n (R n+k ). Classically, as a homogeneous space, it is defined to be the quotient G n (R n+k ) := O(n + k)/(O(n) × O(k)), where O(n) denotes the orthogonal group. The real oriented Grassmann manifold of all oriented n dimensional linear subspaces in the Euclidean space R n+k is denoted by  G n (R n+k ), and is defined to be  G n (R n+k ) := SO(n + k)/(SO(n) × SO(k)), Date : June 15, 2018. The research by Djordje Baralić leading to these results has received funding from the Grant 174020 of the Ministry for Education and Science of the Republic of Serbia. The research by Pavle V. M. Blagojević leading to these results has received funding from DFG via the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”, and the grant ON 174008 of the Serbian Ministry of Education and Science. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1440140 while Pavle V. M. Blagojević was in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall of 2017. The research by Roman Karasev leading to these results has received funding from the Federal professorship program grant 1.456.2016/1.4, the Russian Science Foundation grant 18-11-00073, and the Russian Foundation for Basic Research grant 18-01-00036. 1 arXiv:1709.10492v2 [math.AT] 14 Jun 2018