Properties of continuous functions on a normed space and its sphere

Known well is the problem of finding configurations of points of the Euclidean sphere $S^n$ that can be put to one level of any continuous function on $S^n$ by a rotation of $S^n$. The paper is devoted to various ways of transferring this problem to the case of a normed space. Here is one of the results. Let $f$ and $g$ be two even continuous functions on an $n$-dimensional normed space $E$, and let $f(0)<f(x)$ for each nonzero $x\in E$. Then $E$ contains $n$ unit vectors $e_1,\dots,e_n$ such that for any $1\le i<j\le n$ we have $f(e_i+e_j)=f(e_i-e_j)$ and $g(e_i+e_j)=g(e_i-e_j)$. Bibl. – 16 titles.