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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?
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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.

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