Abstract
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 \(E_n \subset E\) of size n such that \(E_n \rightarrow E\) 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.
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The author would like to thank Philippe Rigollet and the anonymous reviewers for their positive assessments and constructive comments, which greatly helped to improve the paper.
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Gouic, T.L. Recovering a Metric from Its Full Ordinal Information. Discrete Comput Geom 69, 123–138 (2023). https://doi.org/10.1007/s00454-022-00452-2
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DOI: https://doi.org/10.1007/s00454-022-00452-2