A Measure of Hyperconvexity for a Complete Ultrametric Space

Abstract

It has been shown that an ultrametric space is spherically complete if and only if it is hyperconvex. Furthermore, every spherically complete ultrametric space is complete. This, in turn, means that every hyperconvex ultrametric space is complete. However, the converse of the previous statement is not true. In this note, we show that for an ultrametric space X, its ultrametrically injective hull denoted by T_X is spherically complete. Furthermore, we study the conditions under which a complete ultrametric space will be hyperconvex. We will do this by introducing a numerical parameter that enables us to measure the lack of hyperconvexity in a complete ultrametric space.The advantage of the proposed method is that given any complete ultrametric space, we can use the method to deduce if the space is hyperconvex or not.