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This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.

Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.

There is some info in an earlier MO questionan earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:

         
          (From p.335 of conference version.)

This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.

Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.

There is some info in an earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:

         
          (From p.335 of conference version.)

This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.

Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.

There is some info in an earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:

         
          (From p.335 of conference version.)
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Joseph O'Rourke
Joseph O'Rourke
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This is, in general, a difficult problem, addressed in the paper below. It goes under the phrase: the beltway reconstruction problem.

Lemke, Paul, Steven S. Skiena, and Warren D. Smith. "Reconstructing sets from interpoint distances." Discrete & Computational Geometry. Springer Berlin Heidelberg, 2003. 597-631.

There is some info in an earlier MO question. Sets with nonunique reconstructions are called homometric. Here is a snippet from the above paper, addressing a property of regular polygons:

         
          (From p.335 of conference version.)