LemmaStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A compact convex set and an exterior point admit a complex-linear separator

Statement

Let $K \subseteq C^{m}$ be a nonempty compact convex set, and let $p \in C^{m} ∖ K$ . Then there is a complex-linear functional

$L (z) = a_{1} z_{1} + \dots + a_{m} z_{m}$

and a real number $β$ such that

$Re L (z) \leq β < Re L (p) (z \in K) .$

Facts & Assumptions

Given: A nonempty compact convex set $K \subseteq C^{m}$ and a point $p \notin K$ .

[L1]

A point outside a nonempty closed convex subset of Euclidean space admits a strict real-linear separating hyperplane (A point outside a nonempty closed convex set is strictly separated from it).

Proof

technique · direct

1.1

View $C^{m}$ as $R^{2 m}$ by writing $z_{j} = x_{j} + i y_{j}$ . Since $K$ is compact, it is closed, so [L1] gives real numbers $α_{1}, \dots, α_{m}, β_{1}, \dots, β_{m}$ and a real number $β$ such that $\sum_{j = 1}^{m} α_{j} x_{j} + \sum_{j = 1}^{m} β_{j} y_{j} \leq β < \sum_{j = 1}^{m} α_{j} Re p_{j} + \sum_{j = 1}^{m} β_{j} Im p_{j}$ for every $z = (z_{1}, \dots, z_{m}) \in K$ .

L1given

2.1

Define $a_{j} : = α_{j} - i β_{j}$ and $L (z) : = \sum_{j = 1}^{m} a_{j} z_{j}$ . Then $L$ is complex-linear and $Re L (z) = \sum_{j = 1}^{m} α_{j} Re z_{j} + \sum_{j = 1}^{m} β_{j} Im z_{j} .$ Substituting this identity into step 1.1 gives the stated strict separation.

step 1.1algebra∎

Depends on

A point outside a nonempty closed convex set is strictly separated from it

Used by

Convex domains are holomorphically convex Theorem

Dependency tree · two levels

5 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)