Before revision:  In addition, we prove that for every metric space (X,d) it holds that \sqrt{2}-1\le\beta^*_{(X,d)}, and show that there exist a metric space for which \beta^*_{(X,d)}\le \frac12.
After revision:  In addition, we prove that for every metric space (X,d) it holds that \sqrt{2}-1\le\beta^*_{(X,d)}, and show that there exists a metric space for which \beta^*_{(X,d)}\le \frac12.