Consider $n$ points $\{p_i\}_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances, $$J=\sum_{i=1}^n\sum_{j=1}^n \|p_i-p_j\|^a,$$ is maximized? Here, $a$ could be 1 or 2.

The intuition is that the optimal solution is that all the points should distribute evenly on the circle. Any hints about it? Thanks.

Consider $n$ points $\{p_i\}_{i=1}^n$ located inside or on a circle with radius $r$ in the plane. The question is: how to place the $n$ points so that the sum of inter-point distances, $$J=\sum_{i=1}^n\sum_{j=1}^n \|p_i-p_j\|^a,$$ is maximized? Here, $a$ could be 1 or 2.

The intuition is that the optimal solution is that all the points should distribute evenly on the circle. Any hints about it? Thanks.

Another relevant problem is: what if these points are located on the surface of a sphere instead of the plane? In particular, the points should be within a bounded area on a sphere and the distance is the length of the shortest arc connecting the two points instead of Euclidean distance.