[proofplan] The classification combines two previously established results: the [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393), which characterises pseudoconvex Reinhardt domains as those with convex logarithmic image, and the [Domain of Convergence of a Laurent Series](/theorems/3420), which shows that the [domain of convergence of a Laurent series](/theorems/3420) is a logarithmically convex Reinhardt domain and conversely. The three equivalences (pseudoconvex $\iff$ logarithmically convex $\iff$ domain of convergence) are assembled directly from these two results, with no additional work required. [/proofplan] [step:Establish that pseudoconvexity is equivalent to logarithmic convexity for Reinhardt domains] 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 $\mathbb{R}^n$, where \begin{align*} \operatorname{Log}: (\mathbb{C}^*)^n &\to \mathbb{R}^n \\ (z_1, \dots, z_n) &\mapsto (\log|z_1|, \dots, \log|z_n|). \end{align*} This gives the first equivalence in the theorem statement: $\Omega$ is pseudoconvex if and only if its logarithmic image is convex. [guided] The [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393) provides both directions: **Forward direction** (logarithmic convexity $\implies$ pseudoconvexity): If $\operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n)$ is convex, one constructs a plurisubharmonic exhaustion function on $\Omega$ using the support function of the convex set. Linear functionals $h(s) = a_1 s_1 + \cdots + a_n s_n$ on $\mathbb{R}^n$ pull back to pluriharmonic functions $\operatorname{Re}(a_1 \log z_1 + \cdots + a_n \log z_n)$ on $(\mathbb{C}^*)^n$, and the convex set is the intersection of half-spaces defined by such functionals. The resulting exhaustion witnesses pseudoconvexity. **Reverse direction** (pseudoconvexity $\implies$ logarithmic convexity): If $\Omega$ is pseudoconvex, the psh exhaustion and the torus invariance of Reinhardt domains combine to show that the logarithmic image is convex, via the maximum principle for subharmonic functions applied to holomorphic curves connecting two points in logarithmic coordinates. [/guided] [/step] [step:Connect logarithmic convexity to domains of convergence of Laurent series] By the [Domain of Convergence of a Laurent Series](/theorems/3420): (a) The domain of convergence of any [Laurent series](/page/Laurent%20Series) $\sum_{\alpha \in \mathbb{Z}^n} a_\alpha z^\alpha$ is a Reinhardt domain whose logarithmic image is convex (or empty). (b) Conversely, every Reinhardt domain with convex logarithmic image arises as the domain of convergence of some [Laurent series](/page/Laurent%20Series). Combining with the first step: a Reinhardt domain $\Omega$ is pseudoconvex if and only if it is logarithmically convex, which holds if and only if $\Omega$ is the [domain of convergence of a Laurent series](/theorems/3420). [guided] The [Domain of Convergence of a Laurent Series](/theorems/3420) provides the second equivalence. The key insight is that the condition $|a_\alpha| r^\alpha \to 0$ (necessary and sufficient for absolute convergence of the [Laurent series](/page/Laurent%20Series) at points with $|z_j| = r_j$) becomes linear in logarithmic coordinates: setting $s_j = \log r_j$, the condition $\log|a_\alpha| + \alpha \cdot s \to -\infty$ cuts out a convex set in $\mathbb{R}^n$ (an intersection of half-spaces). Conversely, given any convex [open set](/page/Open%20Set) $V \subset \mathbb{R}^n$, one can construct a [Laurent series](/page/Laurent%20Series) with $V$ as its logarithmic domain of convergence by choosing coefficients $a_\alpha$ adapted to the support function of $V$. This also explains the "moreover" clause about Laurent representability: by the [Laurent Series in Complete Reinhardt Domains](/theorems/3419), every [holomorphic function](/page/Holomorphic%20Function) on a complete Reinhardt domain admits a Laurent expansion convergent in the interior. For a pseudoconvex (hence logarithmically convex) Reinhardt domain, the domain of convergence of this Laurent expansion contains the original domain, giving the representability statement. [/guided] [/step] [step:Assemble the complete classification] Combining the two equivalences, we obtain the full chain for a Reinhardt domain $\Omega \subset \mathbb{C}^n$: \begin{align*} \Omega \text{ is pseudoconvex} &\iff \operatorname{Log}(\Omega \cap (\mathbb{C}^*)^n) \text{ is convex in } \mathbb{R}^n \\ &\iff \Omega \text{ is the domain of convergence of a Laurent series.} \end{align*} The first equivalence is the [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393). The second equivalence is the [Domain of Convergence of a Laurent Series](/theorems/3420). For the Laurent representability statement: let $f \in \mathcal{O}(\Omega)$ where $\Omega$ is a pseudoconvex Reinhardt domain. By the [Laurent Series in Complete Reinhardt Domains](/theorems/3419), $f$ admits a Laurent expansion $f(z) = \sum_{\alpha \in \mathbb{Z}^n} a_\alpha z^\alpha$ converging on $\Omega \cap (\mathbb{C}^*)^n$. The domain of convergence of this series is a logarithmically convex Reinhardt domain $D$ containing $\Omega \cap (\mathbb{C}^*)^n$. By the [Identity Principle](/theorems/3357), the [Laurent series](/page/Laurent%20Series) converges to $f$ throughout $\Omega$. For the converse direction: if $D$ is the [domain of convergence of a Laurent series](/theorems/3420), then $\operatorname{Log}(D \cap (\mathbb{C}^*)^n)$ is convex by the [Domain of Convergence of a Laurent Series](/theorems/3420), so $D$ is pseudoconvex by the [Reinhardt Domain Pseudoconvexity Criterion](/theorems/3393). This completes the classification. [/step]