  In this paper, we prove that the total expected curvature for random spatial
equilateral quadrilaterals with diameter at most $r$ decreases as $r$
increases. To do so, we prove several curvature monotonicity inequalities and
stochastic ordering lemmas in terms the of the action-angle coordinates. Using
these, we can use Baddeley's extension of Crofton's differential equation to
show that the derivative of the expected total curvature is non-positive.
