DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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.

Supporting and strictly separating hyperplanes in Euclidean space

Definition

Let $a \neq 0$ in $R^{n}$ and $b \in R$ . The set

$H (a, b) = {x \in R^{n} : ⟨ a, x ⟩ = b}$

is an affine hyperplane, with inner product as in The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ . It supports a set $C$ at $p \in C$ when $p \in H (a, b)$ and either $⟨ a, z ⟩ \leq b$ for every $z \in C$ or the reverse inequality holds for every $z \in C$ .

A point $x \notin C$ is strictly separated from $C$ when there are $a \neq 0$ and $b$ such that

$⟨ a, z ⟩ \leq b < ⟨ a, x ⟩ (z \in C) .$

Two nonempty sets $C, D$ are separated by a hyperplane when some $a \neq 0$ satisfies $⟨ a, c ⟩ \leq ⟨ a, d ⟩$ for every $c \in C$ and $d \in D$ . No convexity is part of these definitions; it is a hypothesis of the existence theorems below (A convex subset of $R^{m}$ contains every line segment between two of its points).

Depends on

Used by

Dependency tree · two levels

16 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

S. Boyd and L. Vandenberghe, Convex Optimization, §2.5 (standard reference, not scraped)
D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 6 (standard reference, not scraped)