I am looking for a proof of the inequality as follows:



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> > Let $A\_1A\_2....A\_n$ be the regular polygon incribed in a circle $(O)$ with radius $R$. Let $B\_1B\_2....B\_n$ be a polygon incribed the circle $(O)$. We let $x\_{ij}=A\_iA\_j$ and $y\_{ij}=B\_{i}B\_{j}$. Let $f(x)=x^m$ (where $m=1, m=2$), I conjecture that:
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> > $$\sum\_{i<j} f(x\_{ij}) \ge \sum\_{i<j} f(y\_{ij})$$
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The case $ m = 1 $ was proved in our paper in [here](https://ijgeometry.com/wp-content/uploads/2017/07/1.pdf)


**Example:** 


* $n=3$, let $ABC$ be a triangle with sidelength $a, b, c$ then we have the inequality as follows:


$$a^2+b^2+c^2 \le (\sqrt{3}R)^2+(\sqrt{3}R)^2+(\sqrt{3}R)^2=9R^2$$


and


$$a+b+c \le 3\sqrt{3}R$$


* $n=4$, let $ABCD$ be a cyclic quadrilateral with sidelength $a=AB$, $b=BC$, $c=CD$, $d=DA$, $e=AC$, $f=BD$ then we have the inequality as follows:


$$a^2+b^2+c^2+d^2+e^2+f^2 \le (\sqrt{2}R)^2+(\sqrt{2}R)^2+(\sqrt{2}R)^2+(\sqrt{2}R)^2+(2R)^2+(2R)^2=16R^2$$


and 


$$a+b+c+d+e+f \le 4(\sqrt{2}+1)R$$


**See also:**


* [Strengthened version of Isoperimetric inequality with n-polygon](https://mathoverflow.net/questions/299056/strengthened-version-of-isoperimetric-inequality-with-n-polygon)



An inequality of a cyclic polygon
As for the sum of squares (m=2), denoting the vectors $\overline{OB\_i}=b\_i$ we get $\sum\_{i<j} y\_{ij}^2=\sum\_{i<j} (b\_i-b\_j)^2=n\sum\_i b\_i^2-(\sum b\_i)^2=n^2R^2-(\sum b\_i)^2$ that is maximal if and only if $\sum b\_i=0$ --- so, in particular, for a regular polygon.


