（z）凸多边形的一个不等式

realnumber

广东广州程汉波(287----79)  21:44:08Erdos-Mordell不等式的推广,应该历史上很早就有人弄过的

whtsg

已知的结果，见BOTTEMA著《几何不等式》16.9

kuing

回复 2# whtsg

谢谢楼上，果然是已知结果，那天晚上我也觉得是被研究过的，可惜查来查去没查到，原来就在《几何不等式》里有提及。

hbghlyj

An optimal inequality Zhiqin Lu
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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.