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Oct 21, 2018 at 2:20 comment added Louis The Boutin-Kemper result shows that, under the same genericity hypothesis as the preprint you link to, the unordered set of distances from a complete graph determine the configuration. In fact, many fewer pairwise distances suffice generically, as shown in the the preprint arxiv.org/abs/1806.08688
Sep 13, 2017 at 19:47 history edited j.c. CC BY-SA 3.0
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Sep 13, 2017 at 19:27 history edited j.c. CC BY-SA 3.0
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Sep 13, 2017 at 19:23 comment added j.c. Also note the mention of this MO question mathoverflow.net/questions/259664 in section 6.3 on the linear automorphisms of the "unsquared measurement variety".
Sep 13, 2017 at 19:21 history answered j.c. CC BY-SA 3.0