Metrically round and sleek metric spaces (Q2692657)

A metric is said to be round if the closure of every open ball is the corresponding closed ball. The authors define a sleek metric as the one for which the interior of every closed ball is the corresponding open ball. A metrizable space whose topology is induced by a sleek (round) metric is called a sleek (round) metric space. The authors have found the following characterizations of non-round metrics and non-sleeks ones. \textbf{Theorem 1}. Let \((X, d)\) be a metric space. The metric space is not round if and only if there is an open set \(U\) and a point \(x \in X \setminus U\) such that the mapping \(d(x, \cdot) \colon U \to \mathbb{R}\) has a minimum. \textbf{Theorem 2}. Let \((X, d)\) be a metric space having at least two points. The metric \(d\) is not sleek if and only if there is \(x \in X\) and an open set \(U\) containing \(x\) such that the mapping \(d(x, \cdot) \colon U \to \mathbb{R}\) has a maximum. The paper contains also some interesting results connected with convexity in metric linear spaces and metric products. In particular, it is shown that a countable product of sleek metric spaces is sleek.