Alphabeta MathHow 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 is star-shaped with respect to when
The point is a star centre. Every convex open set of A convex subset of 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
Used by
- A C¹ field with vanishing curl on a star-shaped open subset of ℝ³ is conservative Corollary
- Cauchy's theorem on a convex complex domain Corollary
- On a star-shaped open domain, closed, exact, conservative, path-independent, and zero-loop are equivalent Corollary
- The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
- A connected complex domain need not be star-shaped Counterexample
- A curl-free C¹ field on the complement of a line that is not conservative Counterexample
- Balls, polydiscs and the distinguished boundary in ℂᵐ Definition
- Radial contraction of a star shaped domain Definition
- A nonvanishing holomorphic function on a disc has a holomorphic logarithm Lemma
- Star-shaped plane domains are homologically simply connected Proposition
- Complex star-shaped and convex domains are the published Euclidean notions under the identification ℂ=ℝ² Remark
- A divergence-free C¹ field on a star-shaped open subset of ℝ³ has a vector potential Theorem
- Poincare's lemma on a star-shaped domain: every closed C1 field is exact Theorem
Dependency tree · two levels
3 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
- J.-B. Campesato, Poincare Lemma, sections 1 and 2 (standard reference, not scraped)