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Question: How is the supporting function of a convex set on $\mathbb{C}$, or $\mathbb{C}^n$, defined?

I seem to not be able to find anything on this. Wikipedia has the case for $\mathbb{R}^n$, but so far I haven't come across anything for sets of complex vectors.

Plue
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  • What makes you think the support function is a thing in the complex setting? Have you run across it in a textbook or paper? – Theo Bendit Nov 01 '18 at 03:09
  • Yes actually. Page 12, here: https://ac.els-cdn.com/S0022247X97951830/1-s2.0-S0022247X97951830-main.pdf?_tid=db610b88-a038-4a01-b887-037b14d10267&acdnat=1541042099_b2a0a20c9028528691c605d7e2469d8f – Plue Nov 01 '18 at 03:13
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    Every complex vector space is also a real Hilbert space, by taking the real part of the inner product. Now proceed as you would in the real setting. – max_zorn Nov 01 '18 at 04:06

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