Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04
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.
The geometric simplex spanned by affinely independent vertices
Definition
Let be affinely independent. The geometric simplex spanned by these vertices is
The numbers are the barycentric coordinates of the point. When , the simplex is the single point .
Used by
- The geometric realization of an abstract simplicial complex Definition
- Barycentric coordinates are unique Lemma
- Crystallographic alcove diagrams: the affine list realized by Weyl types A–G Lemma
- Euclidean simplices with the same facet-normal Gram matrix are similar facet to facet Lemma
- Generic galleries, boundary-fixed disks, and the gallery-move calculus Lemma
- Highest-root dominance and the fundamental alcove Lemma
- Point stabilizers, vertex residues, and rank-two boundary words Lemma
- The affine slice: faithful isometric action, the alcove simplex, and its facet reflections Lemma
- The boundary of a convex set is Lebesgue null Lemma
- Alcove transitivity, the affine Coxeter presentation, and the length function Theorem
- Classification of affine Coxeter diagrams and their Euclidean simplex realization Theorem
Dependency tree · 0 levels
Nothing. This result depends on no other item in the library.
Sources
- Allen Hatcher, Algebraic Topology (standard reference, not scraped)
- Vidit Nanda, Computational Algebraic Topology, Lecture 01: Complexes (standard reference, not scraped)