|
|
An optimal inequality Zhiqin Lu
An optimal inequality.pdf
(173.76 KB, Downloads: 58)
...
We are interested in the generalisation of the Erdös-Mordell inequality to $n$-polygons. The following inequality is true for any positive number $n$ :
Let $a_1+\ldots+a_n$ be $n$ positive numbers. Then we have
\[
a_1+\ldots+a_n \geqslant \sqrt{a_1 a_2}+\ldots+\sqrt{a_{n-1} a_n}+\sqrt{a_n a_1} \text {. }
\]
In general, we conjecture that
\[
a_1+ \ldots+ a_n \geqslant \sec \frac{\pi}{n}\left(\sqrt{a_1 a_2} \cos \alpha_1+\ldots+\sqrt{a_{n-1} a_n} \cos \alpha_{n-1}+\sqrt{a_n a_1} \cos \alpha_n\right)
\]
for $\alpha_1+\ldots+\alpha_n=\pi$.
It is not hard to see that the above inequality, if true, would give the generalisation of the Erdös-Mordell inequality to $n$-polygons. |
|
whtsg
Posted 2014-3-18 10:17
hbghlyj
Posted 2022-11-8 07:45
