By definition the $q$-extent of a metric space $X$ is the maximum average distance between $q$ points; i.e. $$xt_q(X):=\max_{x_1,\dots,x_q}xt_q(x_1 ,\dots ,x_q)$$ where $$xt_q(x_1 ,\dots,x_q) = {q\choose 2}^{-1}\sum_{i<j} \operatorname{dist}(x_i , x_j) .$$
The following image is a screenshot of a lecture. There the lecturer calculated the $q$-extent of round sphere which I think it is wrong. because for $q=4$ it fails. What is the correct solution?
I think it should be $2\pi/4=\pi/2$ but how?
Update: After a few search, I found the following example which shows that my claim is wrong. But I don't understand the author argument. (Or I misunderstood the concept of $q$-extent!! For example, my thought of $4$-extent of $\Bbb S^1$ is that to reach the maximum amount we should put them pair of antipodal points, but this is wrong.)
Q: What does $q$-extent of $\Bbb S^1$ represent exactly? I know that for $q=2$ it is "diameter".
