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?