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!



Constructing a polygon of $n$ facets from a set of positive values representing the length of the facets
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.


