CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedaudited 2026-09-06

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.

Two closed convex sets can have no strong separator

Statement refuted

Two disjoint closed convex sets in a normed space always admit a strong separator.

Facts & Assumptions

Given: $A = R \times {0}$ and $B = {(s, t) : t \geq e^{- s}}$ in $R^{2}$ .

[F1]

The exponential is continuous and convex (A function differentiable at $c$ is continuous at $c$ , The two-point convexity inequality for the exponential function).

Counterexample

technique · direct

1.1

$A$ is closed convex. By [F1], $B$ is the closed epigraph of a convex function, hence closed and convex; it is disjoint from $A$ because $e^{- s} > 0$ .

givenF1

2.1

The points $(n, 0) \in A$ and $(n, e^{- n}) \in B$ have distance $e^{- n} \to 0$ . A strong gap $β - α > 0$ for $f$ would imply $∥ a - b ∥ \geq (β - α) / ∥ f ∥$ for all $a \in A, b \in B$ , impossible.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Gerald Teschl, Topics in Real and Functional Analysis, Problem 5.2 (standard reference, not scraped)