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On reconstructing $n$-point configurations from the distribution of distances or areas.
Gregor Kemper and Mireille Boutin
One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between the points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations. In the second part of the paper, the distribution of aread of subtriangles is used for characterizing point configurations. Again it turns out that most configurations are reconstructible from the distribution of areas, though there are counterexamples.