Умови екстремальності допустимого елемента для узагальненої задачі Штейнера в деякому полінормованому просторі

Abstract

It is known that an important extremal problem in a linear normed space is the classic problemofSteiner, 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 findingin the setof the linear normed space of such a point, that the sum of the weight distance from several fixed points of this space to this pointwould be minimal[2, p. 468;3;4].The generalizedSteiner’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 corre-sponding 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 approxi-mation 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. Korney-chuk [7], O.I. Stepants [8,9] and others.In this article the conditions of extremalof an allowable element for the generalized problem of steiner in polynormed space, which generalize the re-sults obtained, in particular for the above special cases are established