TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-28

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Convex domains are holomorphically convex

Statement

Let $Ω \subseteq C^{m}$ be a convex domain, and let $K ⋐ Ω$ be compact. Then

${\hat{K}}_{Ω} \subseteq conv (K) .$

In particular, $Ω$ is holomorphically convex.

Facts & Assumptions

Given: A convex domain $Ω \subseteq C^{m}$ and a compact set $K ⋐ Ω$ .

[L1]

A point outside a compact convex set can be strictly separated from it by the real part of a complex-linear functional (A compact convex set and an exterior point admit a complex-linear separator).

[L2]

The holomorphic hull is defined by inequalities against all holomorphic functions on $Ω$ (Holomorphic hulls and holomorphic convexity).

[L3]

Convex subsets contain the line segment between any two of their points (A convex subset of $R^{m}$ contains every line segment between two of its points).

Proof

technique · direct

1.1L1L2given

Let $p \in Ω ∖ conv (K)$ . Since $conv (K)$ is compact and convex, [L1] gives a complex-linear functional $L$ such that $Re L (z) \leq β < Re L (p)$ for every $z \in conv (K)$ , hence in particular for every $z \in K$ . The holomorphic function $h (z) : = \exp (L (z))$ then satisfies $∣ h (z) ∣ \leq e^{β} < ∣ h (p) ∣$ on $K$ . By [L2], this excludes $p$ from ${\hat{K}}_{Ω}$ .

2.1L3step 1.1∎

Step 1.1 proves ${\hat{K}}_{Ω} \subseteq conv (K)$ . Because $Ω$ is convex, [L3] gives $conv (K) \subseteq Ω$ . In finite-dimensional Euclidean space the convex hull of a compact set is compact, so ${\hat{K}}_{Ω}$ is contained in a compact subset of $Ω$ . Therefore ${\hat{K}}_{Ω} ⋐ Ω$ , and $Ω$ is holomorphically convex.

Depends on

Used by

Convex domains are domains of holomorphy Corollary
The bidisc is holomorphically convex Example

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Jiří Lebl, Tasty Bits of Several Complex Variables, Exercise 2.1.7 (standard reference, not scraped)
Harold P. Boas, Lecture Notes on Several Complex Variables, Example 12 (standard reference, not scraped)