Geodesic
Geometry of finite-dimensional normed spaces and continuous functions on the Euclidean sphere
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 107-117 
V. V. Makeev. Geometry of finite-dimensional normed spaces and continuous functions on the Euclidean sphere. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 107-117. http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a9/
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     author = {V. V. Makeev},
     title = {Geometry of finite-dimensional normed spaces and continuous functions on the {Euclidean} sphere},
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     year = {2005},
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     url = {http://geodesic.mathdoc.fr/item/ZNSL_2005_329_a9/}
}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

Let $\mathbb R^n$ be the $n$-dimensional Euclidean space, and let $\|\cdot\|$ be a norm in $\mathbb R^n$. Two lines $\ell_1$ and $\ell_2$ in $\mathbb R^n$ are said to be $\|\cdot\|$-orthogonal if their $\|\cdot\|$-unit directional vectors $\mathbf e_1$ and $\mathbf e_2$ satisfy $\|\mathbf e_1+\mathbf e_2\|=\|\mathbf e_1-\mathbf e_2\|$. It is proved that for any two norms $\|\cdot\|$ and $\|\cdot\|'$ in $\mathbb R^n$ there are $n$ lines $\ell_1,\ldots,\ell_n$ that are $\|\cdot\|$- and $\|\cdot\|'$-orthogonal simultaneously. Let $f\colon S^{n-1}\to\mathbb R$ be a continuous function on the unit sphere $S^{n-1}\subset \mathbb R^n$ with center $O$. It is proved that there exists an $(n-1)$-cube $C$ centered at $O$, inscribed in $S^{n-1}$, and such that all sums of values of $f$ at the vertices of $(n-3)$-faces of $C$ are pairwise equal. If the function $f$ is even, then there exists an $n$-cube with the same properties. Furthermore, there exists an orthonormal basis $\mathbf e_1,\ldots,\mathbf e_n$ such that for $1\le i we have $f\left(\frac{\mathbf e_i+\mathbf e_j}{\sqrt 2}\right)=f\left(\frac{\mathbf e_i-\mathbf e_j}{\sqrt2}\right)$.

[1] F. J. Dyson, “Continuous functions defined on spheres”, Ann. Math., 54 (1951), 534–536 | DOI | MR | Zbl

[2] T. Hausel, E. Makai, A. Szűcz, “Inscribing cubes and covering by rhombic dodecahedra via equivariant topology”, Mathematica, 2001

[3] A. Hinrichs, C. Richter, New counterexamples to Knaster's conjecture, Preprint, 2003 | MR

[4] B. S. Kashin, S. J. Szarek, “The Knaster problem and the geometry of high dimensional cubes”, C. R. Acad. Sci. Paris, Ser. 1, 336 (2003), 931–936 | MR | Zbl

[5] V. V. Makeev, “Zadacha Knastera o nepreryvnykh otobrazheniyakh sfery v evklidovo prostranstvo”, Zap. nauchn. semin. LOMI, 167, 1987, 169–178 | MR

[6] V. V. Makeev, “Neskolko teorem o delenii nepreryvno raspredelennoi massy”, Zap. nauch. semin. POMI, 329, 2006, 92–106 | MR | Zbl

[7] R. Rattray, “An antipodal point, orthogonal point theorem”, Ann. Math., 60 (1954), 502–512 | DOI | MR | Zbl