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This may be an already solved question. Let us be given with l lengths AB , BC , … , LM , l being l ≥ 3. What is the condition that we can construct a figure with them? And what is the least dimension n required to do so? Is there a general solution for this question?

My question is not restricted to polygons but also includes polyeders, cells, etc.

김세현
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1 Answers1

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If by "figure" you mean a polygon where $\,M \equiv A\,$ then the necessary and sufficient condition is $\,AB+BC+\ldots+LM \ge 2 \cdot \max(AB, BC, \ldots , LM)\,$, which follows from the triangle inequality. This holds in any dimension $\,n \ge 2\,$, and such a polygon can in fact be constructed in a $2$-dimensional subspace of the $n$-dimensional space once the condition is satisfied.

dxiv
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  • @김세현 That's equivalent to saying that the longest side must be no longer than the sum of the rest. That's a necessary condition as follows directly from the triangle inequality, see for example here or here. – dxiv Jan 29 '18 at 06:37