This is a complex continuation of my previous question. There Iosif Pinelis showed that the so obtained closure from taking the intersection of the preimages of the linear functionals indeed coincides with the (closed) convex hull of the set. I want to ask the analogous question for the complex case and I will use his formulation because I like it conceptually much better than mine :-)
Let $V$ be a finite-dimensional $\mathbb{C}$-vector space, $\dim_\mathbb{C} V \geq 2$, and let $U \subseteq V$ be a subset. I am mostly interested in the case $U$ being a domain (i.e. open and connected), bounded or not, but feel free to consider its closure $\bar{U}$ instead. Define the complex hull (temporary name) of $U$ to be the set $$ \tilde{U} := \bigcap_{\varphi \in V^*} \varphi^{-1}(\varphi(U)), $$ where $V^* = \operatorname{Hom}_\mathbb{C}(V,\mathbb{C})$ denotes the complex dual vector space of $V$. My question is thus:
Can one characterize (set-theoretically, topologically) $\tilde{U}$ in more conventional terms?
I suspect that it may be related to one of the many convexity notions in Several Complex Variables, not only because of the analogy with the real case, but also because $\tilde{U}$ is the natural domain of existence for a certain collection of holomorphic maps (each of which can be extended beyond $\tilde{U}$, but not all in the same way).
Feel free to add more appropriate tags.
PS: Apologies for the "flood" of questions. I have gathered a few over the past year where I could use some clarity and only now have a bit more time on my hands to post them.
