CounterexampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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 closed nonconvex set can have two nearest points

Statement refuted

Every nonempty closed subset of $R$ has a unique nearest point to every input point.

Facts & Assumptions

Given: The false statement above.

[F1]

The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ : on $R$ , the Euclidean distance between $x$ and $y$ is $∣ x - y ∣$ .

[F2]

The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement and Open ball, closed ball and sphere in a metric space: a set is closed when its complement is open; an open ball of radius $r$ about $x$ is the set of points at distance less than $r$ from $x$ .

[F3]

The triangle inequality: $∣ a - c ∣ \geq ∣ a - b ∣ - ∣ b - c ∣$ .

Counterexample

1.1F2F3givenchoose

Let $C = {- 1, 1} \subset R$ . It is nonempty. If $y \notin C$ , put $δ = \min (∣ y - 1 ∣, ∣ y + 1 ∣) > 0$ . For any $z$ with $∣ z - y ∣ < δ / 2$ , [F3] gives $∣ z - 1 ∣ \geq ∣ y - 1 ∣ - ∣ z - y ∣ > δ / 2 > 0$ and $∣ z + 1 ∣ > 0$ , so this ball misses $C$ . Thus $R ∖ C$ is open and $C$ is closed by [F2].

2.1F1step 1.1given∎

For input $y = 0$ , both points of $C$ have distance $1$ by [F1], and these are all the points in $C$ . Hence the nearest-point set is exactly ${- 1, 1}$ , which is not a singleton. The claimed unique projection fails for this nonconvex closed set.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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