Consider $\Omega$ an open set of $\mathbb{C}$ and $f : \Omega \to \mathbb{R}$ a $C_{\infty}$-function. The following two definitions are well-known (at least the first one) but I prefer to recall them :
The function $f$ is subharmonic if for all $z\in \Omega$, there is $r>0$ such that $$f(z) \leq \frac{1}{2\pi}\int_0^{2\pi} f(z+se^{it}) dt,$$ for all $s<r.$
Let us note $j$ the embedding $j : \mathbb{C}\times\mathbb{R} \to P(\mathbb{C}) \times P(\mathbb{R})$, where $P(\mathbb{R})$ is the projective real line and $P(\mathbb{C})$ is the projective complex line.
The function $f$ is $j$-subanalytic if the graph of $f$ $$\text{gr}(f) =\{(x,f(x)) : x\in \Omega\} \subset \mathbb{C}\times\mathbb{R}$$ is a subanalytic set of $P(\mathbb{C}) \times P(\mathbb{R}).$
Does anyone know any link between these two concepts ? For example is a subharmonic function necessarily $j$-subanalytic ? (At least for particular $\Omega$) Any helpful answer or references will be much appreciated.
Edit : The general problem seems very hard, so $\Omega = \mathbb{C}$ would already be an interesting case for me.
