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О геометрической медиане и других медианоподобных точках
The geometric median and some of its generalizations are widely used in economic theory, starting with the works of Wilhelm Launhardt and Alfred Weber on location theory. The most important property of the median of a numerical sample is that the median minimizes the sum of the distance to all elements of the sample. This minimizing property is the basis for determining the geometric median for finite sets of points on the plane. This definition is easily transferred to any metric space, including the Euclidean space . Using integration, the concept of a geometric median extends to bounded submanifolds of any dimension in . There are effective numerical methods for finding the geometric median, but there are no General analytical formulas for calculating it. In this paper, we focus on the geometric medians of bounded domains located in the Euclidean space . The main new results obtained in our work include the conclusion of a new convenient representation of the gradient system for finding the geometric median, as well as the extension of this approach to a wide class of similar optimization problems, where the distance function is replaced by functions of a more general form. It is the solutions to these problems that we call median-like points. They are the closest relatives of the geometric median and are also widely used in modern economic studies.