The Existence Conditions of the Extremal Element for the Generalized Problem of Steiner in Polynormated Space in which the Deviation Between the Elements is Determined with the Help of Sublinear Functionals

• Дата: 2023
• Автори: Гудима, Уляна, Гнатюк, Василь
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2023
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/296428
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

An important place among extremal problems is occupied by the classic Steiner problem, which consists in finding in a given set of linear normed space such a point (Steiner point) to which the sum of the distances from several fixed points of this space will not exceed the sum of the distances from them to any – some other point of the admissible set (will be minimal) [1, p. 314]. In the classic Steiner problem, it is assumed that all segments of the linear normed space are «homogeneous». However, in practice, different «weight» characteristics are attributed to their lengths. As a result, we arrive at the so-called «weighted» Steiner problem [2, p. 468; 3, 4], which, in turn, is a partial case for the problem in which the sum of the distances between fixed points of linear space and points of its set, which were de-termined by weighted norms, were replaced by sums of distances between these points, which, generally speaking, are determined by different norms set on the considered linear space. As a result of this substitution, we ob-tain the generalized Steiner problem in a polynormed space [5]. As you know, there are problems, in particular approximation problems, in which the measure of deviation between fixed elements and elements of a given set is the so-called «distorted metric». The problem considered in the article is obtained as a result of replacing in the generalized Steiner problem in the polynormed space the sum of the distances between fixed points of the linear space and the points of the set of admissible elements, which are determined by various norms given on the linear space, by the sum of the deviations between the specified points, which are determined by by non-negative continuous sublinear functionals defined on the corresponding linear normed spaces. The article establishes some sufficient conditions for the existence of an extremal element (Steiner point) for this problem, which generalize the relevant results obtained, in particular, in [6] for the problem of the best approximation of an element of a linear normed spase by a convex set of this space.