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?
Best Answer
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.