Maximizing the distance sum of some points inside a circle

Question

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.

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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.

Answer 1

Let 0 be the centre of your circle of radius $r$.

If $a=2$, we expand the brackets and write $\sum_{i,j} (p_i-p_j)^2=2n\sum p_i^2-2(\sum p_i)^2\leqslant 2nr^2$, with equality if and only if all points lie on a circle and 0 is a barycentre.

If $a=1$, we again may suppose that they lie on circle (since the function $x\to\|p-x\|$ is convex, the sum of such functions is maximized when all points lie on the boundary of the disc they belong to.) So, they form a convex cyclic $n$-gon, without loss of generality their order is $p_1p_2\ldots p_n$. For each $j=1,2,\ldots,\lceil n/2\rceil$ the cyclic sum of $\|p_i-p_{i+j}\|$ is maximized for the regular polygon. Indeed, if you denote by $2\varphi_i$ the angle measure of the arc $p_ip_{i+1}$ (not containing $p_{i+2}$), so that $\sum \varphi_i=\pi$, you should maximize $\sum \sin (\varphi_{i}+\varphi_{i+1}+\ldots+\varphi_{j-1})$. All arguments of the sine function $\varphi_{i}+\varphi_{i+1}+\ldots+\varphi_{j-1}$ belong to $[0,\pi]$, and their sum is fixed and equals $j\pi$. Since sine is concave on $[0,\pi]$, the sum of sines is maximal when all sums are equal to $\pi j/n$. This is the case for a regular polygon, and already for $j=1$ this holds only for regular polygons.