On the operator of center of distances between the spaces of closed subsets of the real line (Q6614453)

Let \((K([0, 1]), H)\) be the metric space of all nonempty compact subsets of the interval \([0, 1]\) endowed with the Hausdorff metric \(H\). The authors study the continuity properties of the operator \(S \colon (K([0, 1]), H) \to (K([0, 1]), H)\), where, for each \(A \in K([0, 1])\), \(S(A)\) is the center of distances of the compact set \(A\),\N\[\NS(A):= \{\alpha \in [0, \infty) \colon \forall x \in A \ \exists y \in X \ |x - y| = \alpha\}.\N\]\NThe following theorem is the first main result of the paper. \medskip\N\N\textbf{Theorem~1.} The mapping \(S \colon (K([0, 1]), H) \to (K([0, 1]), H)\) is continuous at \(A \in K([0, 1])\) if and only if \(S(A) = \{0\}\).\N\NThe authors give a nontrivial example of the point of continuity of the operator~\(S\). \medskip\N\N\textbf{Theorem~2.} The equality\N\[\NS(\widehat{C}) = \{0\}\N\]\Nholds for the Cantorval\N\[\N\widehat{C} := C \cup \bigcup_{n \in \mathbb{N}} G_{2n-1},\N\]\Nwhere \(G_k\) is the union of all intervals removed in the \(k\)-the step of construction of the Cantor ternary set \(C\).\N\NTo formulate the next result of the paper we recall \medskip\N\N \textbf{Definition.} A mapping \(F \colon (K([0, 1]), H) \to (K([0, 1]), H)\) is semicontinuous at \(A\) if, for every \(\varepsilon > 0\), there is \(\delta > 0\) such that, for every \(Z \in K([0, 1])\) with \(H(Z, A) < \delta\), we have\N\[\NF(Z) \subseteq F(A)_{\varepsilon},\N\]\Nwhere \(F(A)_{\varepsilon}\) is the \(\varepsilon\)-neighborhood of \(F(A)\). \medskip\N\N\textbf{Theorem~3.} The function \(S \colon (K([0, 1]), H) \to (K([0, 1]), H)\) is upper semicontinuous at each \(A \in K([0, 1])\). Moreover, for any \(A \in K([0, 1])\), there is a point \(p \in [0, 1]\) such that the sequence \(S^n(A)_{n \in \mathbb{N}}\) converges to \(\{0, p\}\) in the space \((K([0, 1]), H)\).\N\NThe last theorem implies that \(A \subseteq [0, 1]\) is a fixed point of the operator\N\[\NS \colon (K([0, 1]), H) \to (K([0, 1]), H)\N\]\Nif and only if \(0 \in A\) and \(\operatorname{card}(A) \leqslant 2\).\N\NThe paper contains also interesting results which describe the structure of the center of distances for compact nowhere dense subsets of \([0, 1]\) and for some closed subsets of \([0, \infty)\).