DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-13

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.

Star-shaped open subsets of Euclidean space

Definition

A nonempty open set $U \subseteq R^{n}$ is star-shaped with respect to $a \in U$ when

$a + t (x - a) \in U for every x \in U and 0 \leq t \leq 1.$

The point $a$ is a star centre. Every convex open set of A convex subset of $R^{m}$ contains every line segment between two of its points is star-shaped with respect to each of its points. The nonemptiness and the chosen centre are part of the definition.

Depends on

A convex subset of $\mathbb{R}^m$ contains every line segment between two of its points

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 20 results over 7 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

J.-B. Campesato, Poincare Lemma, sections 1 and 2 (standard reference, not scraped)