# On The Expected Total Curvature of Confined Equilateral Quadrilaterals

## TL;DR

This paper proves that the expected total curvature of random spatial equilateral quadrilaterals decreases as their diameter increases, using curvature inequalities and stochastic ordering in action-angle coordinates.

## Contribution

It introduces new curvature monotonicity inequalities and applies Crofton's differential equation extension to analyze expected total curvature behavior.

## Key findings

- Expected total curvature decreases with increasing diameter
- Established curvature monotonicity inequalities
- Applied stochastic ordering and differential equations

## Abstract

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.