Magic numbers in metric spaces

The main topic of this work is the Gross Theorem and its generalization - the Stadje Theorem. According to the Gross Theorem, for every compact connected metric space (X, d) there exists a unique magic number a(X, d) with the following property: For every finite set K ⊂ X there exists a point y ∈ X such that the average distance from y to K is a(X, d). The Stadje Theorem takes any real-valued continuous symmetric function f on X × X instead of a metric d. In this work we give a proof both of the existence and the uniqueness of the magic number from the Stadje Theorem. Examples of magic numbers in some particular metric spaces are presented. We also study the range of values which can a magic number attain, given some restrictions on the function f or on the space X. 1