Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passverified 2026-09-23 (gpt-6-sol)
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Separation of disjoint convex sets when one is open

Statement

Assume the Axiom of Choice. If U,V⊆X are nonempty disjoint convex sets and U is open, then there is a nonzero f∈X∗ such that

Re⁡f(u)<Re⁡f(v)(u∈U,v∈V).

Facts & Assumptions

Given: The Axiom of Choice and nonempty disjoint convex sets U,V, with U open.

[F1]

An exterior point and a nonempty open convex set admit a strict separating continuous functional (Separate a point from an open convex set).

[A1]

The Axiom of Choice is used through the Hahn–Banach extension in [F1].

Proof

technique · direct
1.1givenalgebra

The difference D=U−V is open and convex, and 0∉D because U∩V=∅.

2.1step 1.1F1A1

Under [A1], apply [F1] to D and 0. It supplies nonzero f∈X∗ with Re⁡f(d)<0 for every d∈D.

3.1step 2.1algebra∎

For u∈U and v∈V, u−v∈D, so Re⁡f(u)−Re⁡f(v)<0. This is the required strict separation.

Depends on

Used by

Dependency tree · two levels

10 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