The Conditions of Extremal of an Allowable Element for the Generalized Problem of Steiner in Some Polynormed Space

• Видавець: Кам'янець-Подільський національний університет імені Івана Огієнка
• Дата: 2022
• Автори: Гудима, Уляна, Гнатюк, Василь
• Формат: Стаття
• Мова: Ukrainian
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2022
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/274034

It is known that an important extremal problem in a linear normed space is the classic problem of Steiner, which consists in finding in the set of this space a point (a point of Steiner) to which the sum of the distances from several fixed points of the space would be minimal [1, p. 314]. In this problem is assumed that all segments of linear normed space are «homogeneous». However, in practice, their lengths often have different «weight» characteristics. We come to the problem of finding in the set of the linear normed space of such a point, that the sum of the weight distance from several fixed points of this space to this point would be minimal [2, p. 468; 3; 4]. The generalized Steiner’s problem in a polynormed space is considered in the article. This problem is obtained as a result of replacing in classic Steiner's problem the sum of the distances from fixed points of linear space to the points of the set of its admissible elements, which are determined by one norm by the sum of the distances from the above-mentioned points with positive weighting coefficients, which are determined by the corresponding different norms determined on this linear space. It is clear that the above extremal problems are special cases of the generalized Steiner’s problem in polynormed space. A special case of this problem is also the problem of the best approximation of an element of a linear normalized space by a convex set of this space, which has been studied by many authors. The main results of research for the problem of the best approximation of an element of a linear normed space are summarized, in particular, in the monographs of N. I. Ahiezer [5], V. K. Dzyadyk [6], M. P. Korneychuk [7], O. I. Stepants [8, 9] and others. In this article the conditions of extremal of an allowable element for the generalized problem of steiner in polynormed space, which generalize the results obtained, in particular for the above special cases are established.