# Polytopes and polyhedral cones in complex Euclidean space

Given $$A \in \mathsf{M}_{m \times n}(\mathbb{R})$$ and $$b \in \mathbb{R}^m$$, the polyhedron with respect to $$A$$ and $$b$$, denoted by $$P(A,b)$$, is defined by $$\{ x \in \mathbb{R}^n \mid Ax \le b \}.$$ If $$P(A,b)$$ is compact, then $$P(A,b)$$ is called a polytope; if $$b=0$$, then $$P(A,b)$$ is called a polyhedral cone.

As is well-known, $$P(A,b)$$ is a polytope if and only if there are vectors $$v_1,\dots,v_k \in \mathbb{R}^n$$ such that $$P(A,b) = \text{conv}(\{v_1,\dots,v_k\})$$. Similary, $$P(A,0)$$ is polyhedral cone if and only if there are vectors $$v_1,\dots,v_k \in \mathbb{R}^n$$ such that $$P(A,0) = \text{coni}(\{v_1,\dots,v_k\})$$.

In $$\mathbb{C}^n$$, it seems reasonable to define a polytope and polyhderal cone as the convex hull and conical hull of a finite collection of vectors, respectively.

The map $$f: \mathbb{C}^n \longrightarrow \mathbb{R}^{2n}$$ defined by $$\begin{bmatrix} z_1 \\ \vdots \\ z_n \end{bmatrix}\in\mathbb{C}^n \longmapsto \begin{bmatrix} \Re z_1 \\ \Im z_1 \\ \vdots \\ \Re z_n \\ \Im z_n \end{bmatrix} \in \mathbb{R}^{2n}$$ is a linear isomorphism (here $$\mathbb{C}^n$$ is viewed as a real, $$2n$$-dimensional vector space). With the definition above, $$V$$ is a polytope (respectively, polyhedral cone) in $$\mathbb{C}^n$$ if and only if $$f(V)$$ is a polytope (respectively, polyhedral cone) in $$\mathbb{R}^{2n}$$.

Questions:

1. Have polytopes/convex polyhedra been defined this way in complex Euclidean space?
2. Has this observation been made before?
3. If so, is there a suitable reference?

One text to consult is Complex Convexity and Analytic Functionals by Andersson, Passare, and Sigurdsson. They give and discuss a number of definitions, including the basic one (a set in $$\mathbb{C}^n$$ is convex if its intersection with any complex line is contractible), which looks like a generalisation of your definition.