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The input of my problem is a set of positive values $a=\{a_1,...,a_n\}$ where $n\geq 3$.

I want to construct an $n$-gon where the lengths of the $n$ facets are the values $a_i$ for $i=1,...,n$.

My questions are:

  • I suppose this problem has been studied before. Is it possible to have some references?
  • From a set $a$, is it always possible to construct a polygon? Are there any conditions on the values $a_i$?

For the second question, if we suppose $a_1$ is the highest value in the set $a$, I guess we must have the condition that $\sum_{i=2}^na_i\geq a_1$. But is there enough?

Thank you very much for your help!

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  • $\begingroup$ As an aside, projecteuclid.org/euclid.jdg/1214457034 discusses the relevant moduli space $\endgroup$
    – Steve Huntsman
    Commented Mar 16, 2016 at 23:55
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    $\begingroup$ Of course it is enough. You may do it by induction: having (convex) $(n-1)$-gon with sides $a_1, a_2,\dots,a_{n-2},a_{n-1}+a_n$ we may replace triangle with sides $a_{n-1}+a_n,a_{n-2}$ and some third size $x$ to a quadrilateral with sides $x,a_{n-2},a_{n-1},a_n$, which is a union of two triangles with sides $a_{n-1}+a_n-\varepsilon,a_{n-2},x$ and $a_{n-1}+a_n-\varepsilon$, $a_{n-1},a_n$. $\endgroup$
    – Fedor Petrov
    Commented Mar 17, 2016 at 0:03
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    $\begingroup$ See the earlier related MO question and answer here, which cites references. The condition you mention allows not only construction of a polygon, but of a convex polygon, and in fact, a triangle. $\endgroup$
    – Joseph O'Rourke
    Commented Mar 17, 2016 at 0:30
  • $\begingroup$ Another result in this direction is that under above necessary condition there exists unique cyclic polygon with given sides (in given order). $\endgroup$
    – Fedor Petrov
    Commented Mar 17, 2016 at 6:41

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No, it is not always possible, because it is not always possible to partition the (cyclic) sequence of edge-length into 3 portions of adjacent edges, for whose sums the triangle inequality holds; e.g. 1,1,3 doesn't allow the construction of a polygon.
Your condition on existence suffices, as has been already demonstrated in the comment of Fedor Petrov.

If a polygon however exists, then there are several ways of making it unique, even in the convex case; you could ask either for minimal or maximal area or, you could demand its corners to be cocyclic, to name just some.

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    $\begingroup$ But maximal side is assumed to be less than sum of other sides. $\endgroup$
    – Fedor Petrov
    Commented Mar 17, 2016 at 7:07
  • $\begingroup$ @FedorPetrov you are right; I overlooked that, so the part related to the non- existence is not relevant for the question. I will have to edit my answer accordingly. $\endgroup$
    – Manfred Weis
    Commented Mar 17, 2016 at 7:29

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