DefinitionDefinition: 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 $v_{0}, \dots, v_{n} \in R^{m}$ be affinely independent. The geometric simplex spanned by these vertices is $[v_{0}, \dots, v_{n}] : = {\sum_{i = 0}^{n} λ_{i} v_{i} : λ_{i} \geq 0, \sum_{i = 0}^{n} λ_{i} = 1} .$

The numbers $λ_{0}, \dots, λ_{n}$ are the barycentric coordinates of the point. When $n = 0$ , the simplex is the single point $v_{0}$ .

Used by

The geometric realization of an abstract simplicial complex Definition
Barycentric coordinates are unique Lemma

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)