PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

Nonnegative combinations, affine precomposition, and finite pointwise maxima preserve convexity

Statement

The following operations preserve convexity on their natural convex domains:

a finite linear combination $\sum_{j} a_{j} f_{j}$ with $a_{j} \geq 0$ ;
precomposition $f \circ A$ with an affine map $A (x) = T x + b$ , where $T$ is Euclidean linear (A linear map $L : R^{m} \to R^{n}$ in Euclidean coordinates).

The pointwise maximum of a nonempty finite family of convex functions on a common convex domain is convex.

Finite sums and maxima use Finite sums and finite products, by recursion, Laws of finite sums and finite products, and Every nonempty finite set of reals has a maximum and a minimum.

Facts & Assumptions

Given: Convex functions on the domains named in the Statement.

[F1]

The function $f : C \to R$ is convex when $f ((1 - t) x + t y) \leq (1 - t) f (x) + t f (y)$ for all $x, y \in C$ and $t \in [0, 1]$ (Convex and strictly convex functions on Euclidean convex sets).

Proof

technique · direct

1.1

For a nonnegative finite combination, multiply the inequality [F1] for $f_{j}$ by $a_{j} \geq 0$ and add over $j$ . This gives the convexity inequality for $\sum_{j} a_{j} f_{j}$ .

F1algebra

1.2

An affine map satisfies $A ((1 - t) x + t y) = (1 - t) A (x) + t A (y)$ . Applying [F1] to the outer function gives convexity of $f \circ A$ on the convex preimage domain.

F1algebra

2.1

Let $g = \max_{j} f_{j}$ . For each $j$ , [F1] gives $f_{j} ((1 - t) x + t y) \leq (1 - t) f_{j} (x) + t f_{j} (y) \leq (1 - t) g (x) + t g (y) .$ Taking the nonempty finite maximum over $j$ proves the inequality for $g$ .

F1algebra∎

Depends on

Used by

A finite maximum of affine functions and its active subgradients Example

Dependency tree · two levels

22 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, §§3.2.1–3.2.3 (standard reference, not scraped)