CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 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.

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Every local minimum of a convex function on an interval is a global minimum

Statement

Every local minimum $c$ of a convex function $f : I \to R$ on an interval is a global minimum: $f (c) \leq f (y)$ for every $y \in I$ .

Facts & Assumptions

Given: A convex $f : I \to R$ on an interval and a local minimum $c \in I$ .

[L1]

A function is convex when the convex-combination inequality holds for every weight in $[0, 1]$ (Convex, strictly convex, concave, strictly concave, and midpoint-convex real functions on an interval).

[L2]

To have a local minimum at $c$ means that some radius $r > 0$ satisfies $f (c) \leq f (x)$ whenever $x \in I$ and $∣ x - c ∣ < r$ (Local (relative) maximum and minimum of $f : A \to R$ at a point, the strict forms, and what it means for the point to be interior to $A$ ).

Proof

technique · contradiction

1.1

Fix $r > 0$ as in [L2], and suppose for contradiction that some $y \in I$ has $f (y) < f (c)$ .

assume-contraL1L2

2.1

Choose $0 < λ < \min {1, r / ∣ y - c ∣}$ when $y \neq c$ and put $z = (1 - λ) c + λ y$ ; then $z \in I$ and $∣ z - c ∣ < r$ .

step 1.1L2algebra

3.1

Convexity gives $f (z) \leq (1 - λ) f (c) + λ f (y) < f (c)$ , contradicting the local-minimum inequality.

step 1.1step 2.1discharge-contradiction∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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, §3.1 (standard reference, not scraped)