I have the following question in Shabat p.59:
Prove that any convex Reinhardt domain is logarithmically convex.
I think I have a good idea about how to show this, but need to be clear on the definitions first, which I'm a little unsure about. Specifically, the concept of a convex domain in $\mathbb{C}^n$. I'm assuming this is simply a domain in $\mathbb{C}^n$ such that for two points in $D \subset \mathbb{C}^n$ given by $z = (z_1,...,z_n), w = (w_1,...,w_n)$, we have the line $ tz + (1-t)w \in D$, for $t \in [0,1]$.
So it remains to show that $ t \text{ ln}|z| + (1-t)\text{ ln}|w| \in \mathbb{R}^n$ for $t\in [0,1]$.
Also, the definition of a Reinhardt domain is the standard one: for $(z_1,...,z_n) \in D$, we have $(z_1e^{i\theta_1},...,z_ne^{i\theta_n}) \in D$.
Is this the correct formulation, or am I way off? Thanks in advance.