A generalization of Erdős–Mordell inequality [closed]

Question

I proposed my conjecture generalization of Erdős–Mordell inequality as following:

Let $A_1A_2....A_n$ be a polygon in a plane, $P$ be the point in $A_1A_2....A_n$. Let $d_i$ be the distances from $P$ to $A_iA_{i+1}$, for $i=1,...,n$ and $A_{i+1}=A_1$. Then:

$$\sum_{1}^{n}{ PA_i} \ge \frac{1}{cos{\frac{\pi}{n}}}\sum_{1}^{n}{d_i} $$

Equality holds when $A_1A_2....A_n$ be the regular polygon, and $P$ is its center.

Answer 1

This has been proved here for convex polygons: Lenhard, H. C., Verallgemeinerung und Verschärfung der Erdös-Mordellschen Ungleichung für Polygone, Arch. Math. 12(1961), 311-314.