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Has there been something written about the following question?

Question: Given a 0/1-polytope, what is the shortest edge lengths with which this polytope can be realized as a 0/1-polytope.

The realization shall respect the angles of the polytope, so not only the combinatorial type.

Example. The regular $n$-crosspolytope can be realized (asymptotically) with edge length $$\ell\ge \sqrt{n/2},$$ but not shorter.

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