We know that the real field is not interpretable in the complex field (with the language of field theory) and the usual way to define convex sets in complex vector spaces is via real coefficients in the interval $[0,1]$. Namely, a set $C$ is convex if, and only if, $$u,v \in C \ \Rightarrow \ \big( \forall t \in [0,1] \big) \, \big[ tu+(1-t)v \in C \big] \, . $$ Where $[0,1]=[0,1] \times \{ 0 \}$ in the Argand's complex plane.
Recently I was trying to define a convex set using only complex coefficients but I failed, and this question naturally popped in my mind:
(1) is a convex set definable?
I am trying to answer this question using just the language(s) of vector spaces and adding minimal more language to that be just the case.
(2) What are minimal language that imply convex sets to be definable?
If this minimal added language yet do not make the real field interpretable in the complex field this would be a maximal interesting answer.
Thank you.