# co.combinatorics – Maximizing the sum of the three sides length of all triangles having as vertices \$n\$ points

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.

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