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

V. V. Makeev

Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 9, Tome 329 (2005), pp. 107-117

Résumé

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 $f\left(\frac{\mathbf e_i+\mathbf e_j}{\sqrt 2}\right)=f\left(\frac{\mathbf e_i-\mathbf e_j}{\sqrt2}\right)$.