Steiner Ratio and Steiner-Gromov Ratio of Gromov-Hausdorff Space

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In the present paper we investigate the metric space M consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset M from a sufficiently small neighborhood of a generic finite metric space, providing M consists of finite metric spaces with the same number of points, each Steiner minimal tree in M connecting M is a minimal filling for M. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of M are equal to 1/2.