Properties of continuous functions on a normed space and its sphere

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Known well is the problem of finding configurations of points on the Euclidean sphere S n that can be put into one level surface 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 ease 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 all nonzero x ∈ E. Then E contains n unit vectors e 1,…,e n such that for any 1 ≤ i ≤ j ≤ n we have f(e i + e j ): f(e i − e j ) and g(e i + e j ): g(e i − e j ). Bibliography: 16 titles.