DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Radial contraction of a star shaped domain

Definition

For an open $U \subseteq R^{n}$ star-shaped about a specified $c \in U$ , the radial contraction is $F : U \times [0, 1] \to U$ , $F (x, t) = c + t (x - c)$ .

Star-shaped open subsets of Euclidean space says exactly that each displayed value lies in $U$ . The coordinate expression is polynomial, so its restriction is smooth up to both endpoints; $F (x, 0) = c$ and $F (x, 1) = x$ . The centre is part of the data, so $U$ is nonempty. For $n = 0$ the unique nonempty domain is a point and the formula is constant.

Source locator

Lee, Theorem 17.14, p.447.

Depends on

Star-shaped open subsets of Euclidean space

Used by

Poincare lemma for differential forms on star shaped domains Theorem

Dependency tree · two levels

2 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

John M. Lee, Introduction to Smooth Manifolds, second edition (standard reference, not scraped)