The Conditions of Extremality of an Admissible Element in the Problem of Finding a Steiner Point of Several Closed Balls of a Certain Polynormed Space with Respect to a Set of this Space Based on the Dual Representation of the Directional Derivative of the Equivalent Best-Approximation Problem

• Datum: 2025
• 1. Verfasser: Гудима, Уляна
• Format: Artikel
• Sprache: Ukrainisch
• Veröffentlicht: Кам'янець-Подільський національний університет імені Івана Огієнка 2025
• Online Zugang: http://mcm-math.kpnu.edu.ua/article/view/345840
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

As is well known (see, for example, [1, p. 47]), the classical Steiner problem in a linear normed space consists in finding, within a given set of this space, a point (the Steiner point) for which the sum of distances to several fixed points of the space is minimal, that is, does not exceed the sum of distances from the given points to any other point of this set. In practice, one often encounters the so-called «weighted» Steiner problems, in which the distances mentioned above are assigned various «weight» characteristics (see, for example, [1, p. 47]). If, in a «weighted» Steiner problem, the «weighted distances» between the fixed points of a linear normed space and the points of its set are replaced by distances generated, generally speaking, by different norms defined on the considered linear space, then we obtain a Steiner problem in a polynormed space, which is a generalization of the «weighted» Steiner problem (see, for example, [2]). In paper [2], for the case when the set of a polynormed space with respect to which the generalized Steiner problem is considered is convex, duality relations and extremality conditions for an admissible solution of this problem are established. These conditions are based on a duality relation that generalizes the known results obtained for the problem of best approximation of an element of a linear normed space by a convex set of this space (see, for example, [3]). The problem considered in the present work is obtained by replacing, in the generalized Steiner problem in a polynormed space, the fixed points of a linear space over the field of real numbers with closed balls determined by the corresponding norms of this space. As distances between the resulting balls and the points of a fixed set of the linear space, we take the Hausdorff distances between them generated by the corresponding norms.