Let $S$ be a circle with unit circumference. Suppose that $n$ random points are chosen independently uniformly from $S$; choosing one arbitrarily as $x_1$, label the rest $x_2, \dots, x_n$ in clockwise order. What is the expected value of $$\sum_{1\le i\le n} max |x_{i}-x_{i-1}|^2 x_{i}-x_{i-1}|$$ (where $x_0$ is interpreted as $x_n$)? Or even more, what is the distribution of this maximum?
(As written, the expression $|x_{i}-x_{i-1}|$ represents distance in the Euclidean plane; I'd prefer to use the distance along the circumference of the circle, which isn't that different when $n$ is large.)