Vol 212, No 1 (2021)

The Fermat-Steiner problem consists in finding all points in a metric space $X$ at which the sum of the distances to fixed points $A_1,…,A_n$ of $X$ attains its minimum value. This problem is studied in the metric space $\mathscr{H}(\mathbb R^m)$ of all nonempty compact subsets of the Euclidean space $\mathbb R^m$, and the $A_i$ are pairwise disjoint finite sets in $\mathbb R^m$. The set of solutions of this problem (which are called Steiner compact sets) falls into different classes in accordance with the distances to the $A_i$. Each class contains an inclusion-greatest element and inclusion-minimal elements (a maximal Steiner compact set and minimal Steiner compact sets, respectively). We find a necessary and sufficient condition for a compact set to be a minimal Steiner compact set in a given class, provide an algorithm for constructing such compact sets and find a sharp estimate for their cardinalities. We also put forward a number of geometric properties of minimal and maximal compact sets. The results obtained can significantly facilitate the solution of specific problems, which is demonstrated by the well-known example of a symmetric set $\{A_1,A_2,A_3\}\subset\mathbb R^2$, for which all Steiner compact sets are asymmetric. The analysis of this case is significantly simplified due to the technique developed. Bibliography 16 titles.

We show that for a nonnegative monotonic sequence $\{c_k\}$ the condition $c_kk\to 0$ is sufficient for the series $\sum_{k=1}^{\infty}c_k\sin k^{\alpha} x$ to converge uniformly on any bounded set for $\alpha\in (0,2)$, and for any odd $\alpha$ it is sufficient for it to converge uniformly on the whole of $\mathbb{R}$. Moreover, the latter assertion still holds if we replace $k^{\alpha}$ by any polynomial in odd powers with rational coefficients. On the other hand, in the case of even $\alpha$ it is necessary that $\sum_{k=1}^{\infty}c_k<\infty$ for the above series to converge at the point $\pi/2$ or at $2\pi/3$. As a consequence, we obtain uniform convergence criteria. Furthermore, the results for natural numbers $\alpha$ remain true for sequences in the more general class $\mathrm{RBVS}$. Bibliography: 17 titles.