Let $n$-regular polygon $X_1X_2\cdots X_n$ with the circumcribed circle $(O)$. Let $n$ points $A_1, A_2,\cdots,A_n$ lie on the circle $(O)$. Let $x_{ij}=X_iX_j$ (for $1 \le i<j \le n) $. Let $a_{ij}=A_iA_j$ (for $1 \le i<j \le n)$.
Now we denote:
$\{ x_1, x_2, \cdots ,x_{\frac{n(n+1)}{2}} \}=\{x_{ij}, 1 \le i < j \le n \},$
$\{ a_1, a_2, \cdots ,a_{\frac{n(n+1)}{2}} \}=\{a_{ij}, 1 \le i < j \le n \}.$
Conjecture that:
$p_k(x_1,x_2, \cdots , x_{\frac{n(n+1)}{2}}) \ge p_k(a_1,a_2, \cdots , a_{\frac{n(n+1)}{2}})$ for all $k=1,2,...,n-1$
$e_k(x_1,x_2, \cdots , x_{\frac{n(n+1)}{2}}) \ge e_k(a_1,a_2, \cdots , a_{\frac{n(n+1)}{2}})$ for all $k=1,2,...,n$
Equality holds iff $A_1A_2 \cdots A_n$ is n-regular polygon.
Here $p_k$ and $e_k$ are the pơer sum and elementary symmetric functions, respectively see Newton's identities.
See also: My previous question
My question: Could you give a proof, or a reference, or a counterexamples?
