The Relation of Duality and the Criterion of the Extremality of an Element for the Problem of Finding the Distance Between Two Convex Sets of Linear Normed Space

• Дата: 2018
• Автори: Гудима, Уляна Василівна, Гнатюк, Василь Олексійович
• Формат: Стаття
• Мова: Українська
• Опубліковано: Кам'янець-Подільський національний університет імені Івана Огієнка 2018
• Онлайн доступ: http://mcm-math.kpnu.edu.ua/article/view/159385
• Назва журналу: Mathematical and computer modelling. Series: Physical and mathematical sciences

P. L. Chebyshov started the conception of the best approximation of a continuous function on a segment using algebraic polynomials of some order at the middle of the XIX century. Later, the notion of the best approximation was considered for the case of general linear normed spaces. Over time, it became clear that a many tasks of best approximation are partial consequence of the problem of the best approximation of an element of a linear normed space by a convex set. This task is called the problem of finding the distance from the given element of the linear normed space to the convex plurality of this space as well. Important questions of the study of this problem are the establishment of the relation of duality and the criterion of the extremality of its element, the specification of this relation and the criterion for some partial cases and their application.M. P. Korniichuk and V. M. Tikhomirov established the general relation of duality and criterion of the extremality of an element for the problem of the best approximation of an element of a linear normed space by a convex set based on the dual interrelation.An important problem, the partial case of which is the problem of the best approximation of an element of a linear normed space by a convex set of this space, is the problem of finding the distance between two convex sets of linear normed space, which is considered in this paper. In the article establishes the relation of duality for this problem, which reduces this problem to the problem of evaluationing the upper bound in the conjugate space. The obtained relation is used to obtain the criterion of the extremality of an element. Thes results are used to find the distance between two bullets and between the bullet and the hyperplane of the linear normed space.