Geodesic
Recovering the metric of a~$CAT(0)$-space by a~diagonal tube
Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 8, Tome 299 (2003), pp. 5-29

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Let $(x,d)$ be a locally compact geodesically complete $CAT(0)$-space of topological dimension $>1$. It is proved that if each geodesic segment in $X$ admits a unique continuation to a complete geodesic, then the metric $d$ is recovered by the diagonal tube $V\subset X\times X$ corresponding to an arbitrary $r>0$. This partly generalizes V. N. Berestovskii's results on A. D. Aleksandrov spaces of negative curvature. 
@article{ZNSL_2003_299_a0,
     author = {P. D. Andreev},
     title = {Recovering the metric of a~$CAT(0)$-space by a~diagonal tube},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {5--29},
     publisher = {mathdoc},
     volume = {299},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2003_299_a0/}
}
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P. D. Andreev. Recovering the metric of a~$CAT(0)$-space by a~diagonal tube. Zapiski Nauchnykh Seminarov POMI, Geometry and topology. Part 8, Tome 299 (2003), pp. 5-29. http://geodesic.mathdoc.fr/item/ZNSL_2003_299_a0/