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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?


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Your condition amounts to $\xi=\rho_1e^{i\theta}$, $\zeta=\rho_2e^{i\theta}$ which is a (real) convex subset of $\mathbb C^2$. – Roland Bacher Apr 6 '11 at 16:59
Sorry, wrong (I forgot to take the real part). – Roland Bacher Apr 6 '11 at 17:24
Counterexample: Take $(1,1+1000i)$ and $(1000-1000i,1)$. Both work. Their midpoint $(1001/2-500i,1+500i)$ is not in your set. – Roland Bacher Apr 6 '11 at 17:31
Indeed. thank you. – nonameisfinetoo Apr 6 '11 at 17:38

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