Given a unit disc containing a set $S_n$ of $ngg 1$ points, let $p_n$ be the average perimeter of a triangle formed by any three distinct points of $S_n$ (i.e., $p_n$ is the ratio between the sum of the three sides length of all triangles having as vertices three distinct points of $S_n$, and the total number $n choose 3$ of these triangles).
Question: How can we prove that the maximum value $p^*_n$ of $p_n$ over all possible set $S_n$, is attained when all $n$ points of $S_n$ are the vertices of a regular n-gon lying on the unit circle? How can we calculate the value of $p^*$?
Note: I found some publications with questions related to optimization problems over all triangles formed by $n$ points. However, either the points are generated uniformly at random, or the maximization function is different from the problem above. Although it is a very simple (and natural, in my opinion) question, I could not find any reference. Besides finding an answer to this problem, I would be glad to read some papers strongly related to this topic, mainly because this is just the first step towards the study of this problem in higher dimensions.