I know that a closed convex set $\Omega\subset \mathbb{R}^n$ is the intersection of the half spaces containing it. Here a half space is of the form $H=\{x\in: \mathbb{R}^n: f(x)\geq0 \}$, where $f: \mathbb{R}^n \longrightarrow \mathbb{R}$ is an affine map. And $f$ is an affine map if there exists a linear map $h: \mathbb{R}^n \longrightarrow \mathbb{R}$ and a fixed vector $c\in \mathbb{R}$, such that $f(x)=h(x)+c$ for every $x\in \mathbb{R}$.
Now I am reading a paper where a closed half space in $\mathbb{C}^n$ is denoted by $H’=\{z\in: \mathbb{C}^n: a(z)\geq0 \}$ for some affine function $$a:z=(z_k)\mapsto b+ \sum \alpha_kz_k+ \sum \overline{\alpha_kz_k},$$ where $b\in\mathbb{R}, \alpha_k\in\mathbb{C}$. Can anyone tell why here $b\in \mathbb{R}$? And why is the expression of $a$ defined as such, I do not see it as a sum of a linear function and a complex scalar. Can anyone suggest a reference where closed convex sets in $\mathbb{C}^n$ are defined and explained this way?
The point is that you need your function $a$ to be real-valued in order for $a(z) \ge 0$ to make sense. So you take the real part of the sum of a linear function and a scalar.
$$ \text{Re}(\beta + \sum_{k} \alpha_k z_k) = \text{Re}(\beta) + \frac{1}{2} \sum_k \alpha_k z_k + \frac{1}{2} \sum_k \overline{\alpha_k z_k}$$