Complex of abstract cubes and median problem

In this paper a special complex Kn of abstract cubes [2, 3], which contains only n-dimensional cubes is examined. The border of this complex is an abstract (n − 1)-dimensional sphere. It is proved that the abstract sphere contains at least one 0dimensional cube, which belongs to exactly n cubes with dimension 1, if the complex is a homogeneous n-dimensional tree. This result allows to solve, in an efficient way, the problem of median for a skeleton of size 1 of the tree with weighted vertices and edges. The algorithm to calculate the median without using any metric is described. The proposed algorithm can be applied with some modifications, for arbitrary complex of abstract cubes. Mathematics subject classification: 18F15, 32Q60, 68R10 Keyword and phrases: Abstract cube, complex, multidimensional homogeneous tree, abstract sphere, median