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The triangle inequality provides a necessary and sufficient condition for three numbers $a_1, a_2, a_3$ to be the lengths of the sides of a triangle; there is no triangle unless each $a_i$ is less than the sum of the other two, and if that condition is satisfied, then there does exist a triangle with those sides.

Given positive real numbers $a_1, a_2, a_3, a_4$, what is a necessary and sufficient condition for these to be the lengths of the sides of a plane quadrilateral?

What is the corresponding condition for the four numbers to be the sides of a convex plane quadrilateral?

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Each side is less than the sum of the other three. That allows both convex and concave plane quadrilaterals.
First make one straight side out of $a_1+a_2$ and form a triangle. Then nudge it out of straight a little either way to get a convex or concave quadrilateral.
Hmm, that won't work if $a_1$ and $a_2$ are the longest sides. Perhaps it is possible if they are the shortest sides.

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