DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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.

Fenchel conjugate, biconjugate, and subdifferential

Definition

Let $f : R^{d} \to \overline{R}$ be proper in the sense of Proper, convex, and lower-semicontinuous extended-real functions on Euclidean space. Its Fenchel conjugate is $f^{*} : R^{d} \to \overline{R}$ ,

$f^{*} (p) : = \sup_{x \in dom f} (⟨ p, x ⟩ - f (x)) .$

The supremum is taken in $\overline{R}$ ; its defining set is nonempty and consists of real numbers, so $f^{*} (p) \in R \cup {+ \infty}$ . The biconjugate is an extended-real expression that may also take $- \infty$ when its defining supremum is over the empty set:

$f^{* *} (x) : = \sup_{{p \in R^{d} : f^{*} (p) < + \infty}} (⟨ p, x ⟩ - f^{*} (p)),$

with the convention $\sup \emptyset = - \infty$ . For $x \in dom f$ , the subdifferential is

$\partial f (x) : = {p \in R^{d} : f (y) \geq f (x) + ⟨ p, y - x ⟩ for every y \in R^{d}};$

set $\partial f (x) = \emptyset$ for $x \notin dom f$ . A vector in $\partial f (x)$ is a subgradient at $x$ . All pairings use The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ .

Fenchel conjugate, biconjugate, and subdifferential

Definition

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