Abstract

Assume that n independent uniformly distributed random points are set on the unit circle. Construct the convex random polygon with vertices in these points. What are the average area and the average perimeter of this polygon? Brown computed the average area several years ago. We compute the average perimeter and obtain quite similar expressions. We also discuss the rate of the convergence of this value to the limit and evaluate the average value of the sum of squares for the sides of the inscribed triangle.

中文翻译：

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关于内接随机多边形的平均周长

摘要

假设在单位圆上设置了n个独立均匀分布的随机点。用这些点的顶点构造凸随机多边形。该多边形的平均面积和平均周长是多少？布朗计算了几年前的平均面积。我们计算平均周长并获得非常相似的表达式。我们还将讨论该值收敛到极限的速率，并评估内切三角形边的平方和的平均值。