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Dual cone and polar cone http://en.wikipedia.org/wiki/Dual_cone_and_polar_cone are defined only on $\mathbb R^n$. Has anyone seen the extension to $\mathbb C^n$? Any references for these?

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    Are you just wondering or you're asking if someone else already has defined such a thing for a purpose? Because there is probably a way to define such a generalization but I've never heard of a purpose for it, though.2011-12-18
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    I am wondering if someone else already has defined such a thing for a purpose.2011-12-19
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    Very good question then, I am curious too. Would love to see an answer ; +1!2011-12-19
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    The definition on Wikipedia automatically generalizes to any inner product space (and $\mathbb{C}$ has of course an inner product)2011-12-19
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    @Fredrik, you mean to take real part of inner product of complex vectors?2011-12-19
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    @Sunni: If $z_1=a_1+ib_1$ and $z_2=a_2+ib_2$, then we define $z_1 \cdot z_2 = a_1 a_2+b_1b_2$. You can check that this gives the absolute value squared when $z_1=z_2$.2011-12-19

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As I see it, the definition in Wikipedia is for any Linear Space. The book by Aliprantis and Border defines polars for any Dual Set, which is even more general. These general definitions are very useful in Convex Optimization, but I've never seen it used on $\mathbb{C}^n$.