# Strengthened version of Isoperimetric inequality with n-polygon

Let $$ABCD$$ be a convex quadrilateral with the lengths $$a, b, c, d$$ and the area $$S$$. The main result in our paper equivalent to:

$$$$a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\sum{(a-b)^2}$$$$

where $$\sum{(a-b)^2}=(a-b)^2+(a-c)^2+(a-d)^2+(b-c)^2+(b-d)^2+(c-d)^2$$ and note that $$\frac{\sqrt{3}-1}{\sqrt{3}}$$ is the best constant for this inequality.

Similarly with the same form above apply to n-polygon, I propose a conjecture that:

Let a convex polygon $$A_1A_2...A_n$$ with the lengths are $$a_1, a_2, ...a_n$$ and area $$S$$ we have:

$$$$\sum_i^n{a_i^2} \ge 4\tan{\frac{\pi}{n}}S + k\sum_{i < j}{(a_i-a_j)^2}$$$$

I guess that $$k=tan{\frac{\pi}{n}}-tan{\frac{\pi}{n+2}}$$

I am looking for a proof of the inequality above.

Note that using Isoperimetric inequality we can prove that:

$$$$\sum_i^n{a_i^2} \ge 4\tan{\frac{\pi}{n}}S$$$$

Case n=3:

$$$$a^{2} + b^{2} + c^{2} \geq 4 \sqrt{3}S+ (a - b)^{2} + (b - c)^{2} + (c - a)^{2}$$$$

Case n=4: our paper, $$k=\frac{\sqrt{3}-1}{\sqrt{3}}$$, $$k=tan{\frac{\pi}{4}}-tan{\frac{\pi}{6}}$$

$$$$a^2+b^2+c^2+d^2 \ge 4S + \frac{\sqrt{3}-1}{\sqrt{3}}\sum{(a-b)^2}$$$$

The conjectured inequality, with $k=\tan{\frac{\pi}{n}}-\tan{\frac{\pi}{n+2}}$, is false for $n=3$. More specifically, the constant factor $k=1$ is optimal in the Hadwiger--Finsler inequality: e.g., consider $a=b=1$ and $c\approx0$.
• I check in long time, but I can not find the optimal formular of $k$ Jul 11, 2018 at 2:25
• So far, we don't even seem to have a good expression for the optimal $k$ covering the cases of $n=3$ and $n=4$. Jul 11, 2018 at 11:44
• I think optimal of $k \approx \tan\frac{\pi}{n}- \tan\frac{\pi}{n+2}$ Jul 11, 2018 at 12:16