Mean gap from a point to its nearest of 19 others

Twenty points $X_1,\dots,X_{20}$ are placed independently and uniformly on a circle. Measured in degrees along the circle, what is the expected arc distance from $X_1$ to whichever other point lies closest to it?

$9^\circ$. The nearest-neighbour distance $L$ exceeds $x$ (radians) when all $19$ other points avoid an arc of length $2x$, probability $(1-x/\pi)^{19}$. Integrating over $[0,\pi]$ gives $\mathbb{E}[L]=\pi/20$, i.e. $9$ degrees.