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

Conjecture: Let $A_1A_2....A_n$ be the regular polygon incribed a circle $(O)$. Let $B_1B_2....B_n$ be a polygon incribed the circle $(O)$ then:

$$\sum_{i<j} A_iA_j^\alpha \ge \sum_{i<j} B_iB_j^\alpha $$

Where $ 1 \leq \alpha \leq 2$.

• The case $ \alpha = 1 $ was proved in our paper in here

• The case $ \alpha = 2 $ was proved in here

Example:

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

$$a^\alpha + b^\alpha+ c^\alpha \leq 3\times 3^{\frac{\alpha}{2}}R^\alpha$$ Where $ 1 \leq \alpha \leq 2$, $R$ is circumradius.

See also: