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 $(O)$. Denote $x_{ij}=A_iA_j$ and $y_{ij}=B_{i}B_{j}$ then:

$$\sum_{i<j} x_{ij}^\alpha \ge \sum_{i<j} y_{ij}^\alpha $$

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:

  • 1
    Dear Mr Dao, I think polygon $B_1B_2...B_n$ need be convex? – Tran Quang Hung Jul 12 at 9:27
  • 1
    @TranQuangHung I don't think that – Đào Thanh Oai Jul 12 at 14:12
  • 1
    That's right Mr Dao, thank you very much. I thought the regular polygon was convex but this is not really. – Tran Quang Hung Jul 12 at 14:47

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