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I have a convex hull enumerated by vertices ${a_1,\cdots, a_n}$ in the complex plane. I am performing the (complex)square root operation over these vertices. How would the image look like after this operation? whether the convexity is still preserved? Does it going to make a difference if we consider only the principal square root?

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  • $\begingroup$ If you start with just one point $-1$ then the convex hull of this is again $\{-1\}$. The square roots of the points in this set consist of two points $i$ and $-i$ so we get a non-convex set. $\endgroup$ – Kavi Rama Murthy Aug 14 at 5:29
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A counterexample: The image of the segment $[-i, i]$ under the (principle value of the) square root is not convex.

It contains the points $z_{1, 2} = \sqrt{\pm i} = \frac{1 \pm i}{2}$, but not the point $\frac 12 (z_1 + z_2) = \frac 12$.

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