2019
DOI: 10.48550/arxiv.1902.06316
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On The Expected Total Curvature of Confined Equilateral Quadrilaterals

Gabriel Khan

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.

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