[proofplan]
We show that a Reinhardt domain $\Omega \subset \mathbb{C}^n$ is a domain of holomorphy if and only if it is logarithmically convex, by combining two equivalences: the [Solution of the Levi Problem](/theorems/3416) (which identifies domains of holomorphy with pseudoconvex domains) and the [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393) (which identifies pseudoconvex Reinhardt domains with logarithmically convex ones).
[/proofplan]
[step:Reduce the domain-of-holomorphy condition to pseudoconvexity]
By the [Solution of the Levi Problem](/theorems/3416), a domain $\Omega \subset \mathbb{C}^n$ is a domain of holomorphy if and only if $\Omega$ is pseudoconvex. This equivalence holds for all domains in $\mathbb{C}^n$ without any symmetry assumption, so it applies in particular to the Reinhardt domain $\Omega$.
[/step]
[step:Reduce pseudoconvexity of Reinhardt domains to logarithmic convexity]
By the [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393), a Reinhardt domain $\Omega \subset \mathbb{C}^n$ is pseudoconvex if and only if $\operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n)$ is convex in $\mathbb{R}^n$. This is precisely the condition of logarithmic convexity.
[/step]
[step:Chain the two equivalences to obtain the characterisation]
Combining the two equivalences:
\begin{align*}
&\Omega \text{ is a domain of holomorphy} \\
&\quad \Longleftrightarrow \quad \Omega \text{ is pseudoconvex} && \text{(Solution of the Levi Problem)} \\
&\quad \Longleftrightarrow \quad \operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n) \text{ is convex in } \mathbb{R}^n && \text{(Reinhardt Pseudoconvexity Criterion)}
\end{align*}
The last condition is the definition of logarithmic convexity for Reinhardt domains. This gives the desired equivalence: $\Omega$ is a domain of holomorphy if and only if $\Omega$ is logarithmically convex.
[guided]
This theorem is a direct consequence of two deep results proved earlier in the course. Let us verify that both equivalences apply and chain them explicitly.
The [Solution of the Levi Problem](/theorems/3416) states: a domain $\Omega \subset \mathbb{C}^n$ is a domain of holomorphy if and only if $\Omega$ is pseudoconvex. The hypothesis is that $\Omega$ is a domain in $\mathbb{C}^n$ (connected [open set](/page/Open%20Set)). Since every Reinhardt domain is a domain in $\mathbb{C}^n$, the equivalence applies:
\begin{align*}
\Omega \text{ is a domain of holomorphy} \quad \Longleftrightarrow \quad \Omega \text{ is pseudoconvex}.
\end{align*}
The [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393) states: a Reinhardt domain $\Omega \subset \mathbb{C}^n$ is pseudoconvex if and only if $\operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n)$ is convex in $\mathbb{R}^n$. The hypothesis is that $\Omega$ is a Reinhardt domain, which is given. Therefore:
\begin{align*}
\Omega \text{ is pseudoconvex} \quad \Longleftrightarrow \quad \operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n) \text{ is convex in } \mathbb{R}^n.
\end{align*}
Chaining the two biconditionals:
\begin{align*}
&\Omega \text{ is a domain of holomorphy} \\
&\quad \Longleftrightarrow \quad \Omega \text{ is pseudoconvex} && \text{(Solution of the Levi Problem)} \\
&\quad \Longleftrightarrow \quad \operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n) \text{ is convex in } \mathbb{R}^n && \text{(Reinhardt Pseudoconvexity Criterion)}
\end{align*}
The last condition is the definition of logarithmic convexity for Reinhardt domains, so $\Omega$ is a domain of holomorphy if and only if $\Omega$ is logarithmically convex. This completes the proof.
The power lies in the combination: the Levi problem solution (which required the full machinery of $\bar\partial$-theory and Hormander's $L^2$ estimates) translates the analytic condition of being a domain of holomorphy into the geometric condition of pseudoconvexity. The Reinhardt pseudoconvexity criterion then translates the complex-analytic condition of pseudoconvexity into the elementary real-geometric condition of convexity of the logarithmic image.
For a general domain in $\mathbb{C}^n$, testing whether a domain is a domain of holomorphy is a difficult problem: one must understand the entire algebra $\mathcal{O}(\Omega)$ and verify that no [holomorphic function](/page/Holomorphic%20Function) extends past the boundary. For Reinhardt domains, this reduces to a computation that can be done with a pencil: draw the set $\{(\log|z_1|, \dots, \log|z_n|) : z \in \Omega,\, z_j \neq 0\}$ in $\mathbb{R}^n$ and check whether it is convex.
[/guided]
[/step]
