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Suppose for $n\geq 3$ we have, $z_1,\ldots ,z_n\in\mathbb{C}$ and $|z_1|=|z_2|=\cdots=|z_n|=1$. Now I need to determine a property $P$ such that the following is true :

$\sum_{i=1}^n z_i=0\Longleftrightarrow z_1,\ldots ,z_n\mbox{ are the vertices of a polygon satisfying } P$

I have solved the cases for $n=3$ and $n=4$, in the first case, $P$ is equilateral triangle and for the second, $P$ is rectangle. But my methods does not generalize for general case. Can we solve it for general $n$ ? At least I would like to know what happens for $n=5$.

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    @lada, that's not even true for $n=4$, where, as OP knows, rectangles work. It gets worse for larger $n$.2012-08-05

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You can get all the solutions for $n=5$, but it isn't pretty.

There must be a sector of size $4\pi/5$ containing at least 3 of the points. By rotating and reflecting, if necessary, we may assume $z_j=e^{2\pi i\theta_j}$ with $\theta_1=0$, $0\le\theta_1\le1/5$, $\theta_1\le\theta_2\le2/5$. Now provided only that $|z_1+z_2+z_3|\le2$ we can find unique $\theta_4\le\theta_5$ to make $\sum^5z_j=0$.

Thinking of $\theta_1$ and $\theta_2$ as parameters, this gives a 2-parameter family of solutions to $z_1+z_2+z_3+z_4+z_5=0,\qquad|z_j|=1$