| Autor: |
Baralić, Djordje, Blagojević, Pavle V. M., Karasev, Roman, Vučić, Aleksandar |
| Rok vydání: |
2017 |
| Předmět: |
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| Zdroj: |
Forum Mathematicum 30:6 (2018), 1539-1572 |
| Druh dokumentu: |
Working Paper |
| DOI: |
10.1515/forum-2018-0058 |
| Popis: |
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 $1\leq i\leq k$. |
| Databáze: |
arXiv |
| Externí odkaz: |
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