DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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.

Local and strict local extrema for scalar fields on Euclidean open sets

Definition

Let $U\subseteq\mathbb R^m$ be open, $a\in U$ , and $f:U\to\mathbb R$ . The point $a$ is a local minimum when some Euclidean neighbourhood $V$ of $a$ satisfies $f(a)\le f(x)$ for every $x\in U\cap V$ ; it is a strict local minimum when the inequality is strict for $x\ne a$ . Local and strict local maxima reverse these inequalities. Euclidean neighbourhoods use 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. A critical point is as in The Hessian matrix and critical points of a scalar field.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Analysis, Convexity, and Optimization (standard reference, not scraped)