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Hi. I have trouble deciding if the set of couples $ \left( \xi, \zeta \right) \in \mathbb{C}^2 $ with $ Re \left( \xi \text{ } \overline{\zeta} \right) > 0 $ is convex. It is a (real) cone, but is it a convex one? If not, could you provide me with a counter-example?

thx.

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  • $\begingroup$ Your condition amounts to $\xi=\rho_1e^{i\theta}$, $\zeta=\rho_2e^{i\theta}$ which is a (real) convex subset of $\mathbb C^2$. $\endgroup$
    – Roland Bacher
    Commented Apr 6, 2011 at 16:59
  • $\begingroup$ Sorry, wrong (I forgot to take the real part). $\endgroup$
    – Roland Bacher
    Commented Apr 6, 2011 at 17:24
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    $\begingroup$ Counterexample: Take $(1,1+1000i)$ and $(1000-1000i,1)$. Both work. Their midpoint $(1001/2-500i,1+500i)$ is not in your set. $\endgroup$
    – Roland Bacher
    Commented Apr 6, 2011 at 17:31

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