On The Expected Total Curvature of Confined Equilateral Quadrilaterals

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.