DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)audited 2026-08-02

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.

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

Definition

A subset $U\subseteq\mathbb R^m$ is convex when, for all $x,y\in U$ and $t\in[0,1]$ (Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length), the point $(1-t)x+ty$ lies in $U$ . Thus the full line segment from $x$ to $y$ remains in $U$ .

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 84 results over 17 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. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)

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