Recovering metric from full ordinal information

06/11/2015

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by   Thibaut Le Gouic, et al.

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Given a geodesic space (E, d), we show that full ordinal knowledge on the metric d-i.e. knowledge of the function D d : (w, x, y, z) → 1 d(w,x)<d(y,z) , determines uniquely-up to a constant factor-the metric d. For a subspace En of n points of E, converging in Hausdorff distance to E, we construct a metric dn on En, based only on the knowledge of D d on En and establish a sharp upper bound of the Gromov-Hausdorff distance between (En, dn) and (E, d).