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Zapiski Nauchnykh Seminarov POMI, 2005, Volume 329, Pages 107–117
(Mi znsl300)
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This article is cited in 2 scientific papers (total in 2 papers)
Geometry of finite-dimensional normed spaces and continuous functions on the Euclidean sphere
V. V. Makeev Saint-Petersburg State University
Abstract:
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 $1\le i<j\le n$ we have $f\left(\frac{\mathbf e_i+\mathbf e_j}{\sqrt 2}\right)=f\left(\frac{\mathbf e_i-\mathbf e_j}{\sqrt2}\right)$.
Received: 01.03.2005
Citation:
V. V. Makeev, “Geometry of finite-dimensional normed spaces and continuous functions on the Euclidean sphere”, Geometry and topology. Part 9, Zap. Nauchn. Sem. POMI, 329, POMI, St. Petersburg, 2005, 107–117; J. Math. Sci. (N. Y.), 140:4 (2007), 558–563
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https://www.mathnet.ru/eng/znsl300 https://www.mathnet.ru/eng/znsl/v329/p107
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Abstract page: | 235 | Full-text PDF : | 62 | References: | 31 |
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