Recovering a Metric from Its Full Ordinal Information

Given a geodesic space ( E , d ), we show that full ordinal information (quadruple comparison of distances) on the metric d determines uniquely—up to a constant factor—the metric d . Moreover, given any sequence {E_n} of subsets E_n ⊂ E of size n such that E_n → E in Hausdorff distance we construct a metric d_n on E_n from only ordinal information on (E_n, d) and prove rates of convergence of (E_n,d_n) to ( E , d ) in Gromov–Hausdorff distance.