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Geometry of finite-dimensional normed spaces,and continuous functions on the Euclidean sphere

Authors:	V V Makeev

Institution:	(1) St.Petersburg State University, St.Petersburg, Russia

Abstract:	Let ℝⁿ be the n-dimensional Euclidean space, and let { · } be a norm in Rⁿ. Two lines ℓ₁ and ℓ₂ in ℝⁿ are said to be { · }-orthogonal if their { · }-unit direction vectors e ₁ and e ₂ satisfy {e ₁ + e ₂} = {e ₁ − e ₂}. It is proved that for any two norms { · } and { · }′ in ℝⁿ there are n lines ℓ₁, ..., ℓ_n that are { · }-and { · }′-orthogonal simultaneously. Let $\mathbb{S}^{n - 1} \to \mathbb{R}$ be a continuous function on the unit sphere $\mathbb{S}^{n - 1} \subset \mathbb{R}^n$ with center O. It is proved that there exists an (n − 1)-cube C centered at O, inscribed in $\mathbb{S}^{n - 1}$ , 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 e ₁, ..., e _n such that for 1 ≤ i ≤ j ≤ n we have $f\left( {\frac{{e_i + e_j }}{{\sqrt 2 }}} \right) = f\left( {\frac{{e_i - e_j }}{{\sqrt 2 }}} \right)$ . Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 107–117.

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