We know that to form a triangle the 3 sides should obey the triangle inequality . So is there any rule to be followed by the sides of $n$-sided convex polygon.
For Eg:-
$1,2,4$ cannot form a triangle so can we tell if we are given $n$ line segments can we make a $n$-sided convex polygon.
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Martin Sleziak
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^But is this condition a sufficient one to make a convex polygon if you are given n sides?? – user82632 Jun 15 '13 at 21:31
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The longest side must be shorter than the sum of the rest.
Hagen von Eitzen
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The shortest distance between two points is a straight line, and the other sides form a path from one endpoint of the longest side to the other, which must have greater length than the length of that side. Must a convex n-gon exist for all sets of side lengths satisfying the above rule? – Angela Pretorius Jun 07 '13 at 12:52
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@Mod Let $A_1..A_n$ be the polygon. Then $A_1A_3 < A_1A_2+A_2A_3$. Now use induction, look at $A_1A_3A_4..A_n$. – N. S. Jun 07 '13 at 12:53
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put all the non-longest in a line segment $L$, which exceeds the length of the longest side, at a certain angle from it. Now using the juncture points in $L$ you can "bend" $L$ outward from the long side until the tip of $L$ is at distance equal to the length of the long side from one end, then rotate to close into a convex figure. – coffeemath Jun 07 '13 at 12:55