ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedverified 2026-10-15 (gpt-6-luna, gpt-6.1-sol, and delegated Codex review/repair agents; see completed run evidence)

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A vertical separator and biconjugation outside the domain

Example

Let $f : R \to {0, + \infty}$ be the indicator of $[0, \infty)$ : $f (y) = 0$ for $y \geq 0$ and $f (y) = + \infty$ for $y < 0$ . At $z = (- 1, 0)$ , the epigraph is separated by a vertical hyperplane (one whose coefficient of the epigraph-height coordinate is zero). The conjugate is $f^{*} (p) = 0$ for $p \leq 0$ and $+ \infty$ for $p > 0$ , so $f^{* *} (- 1) = + \infty = f (- 1)$ . The affine minorants $ℓ_{k} (y) = - k y$ , $k \geq 0$ , display the unbounded dual values at $y = - 1$ .

Facts & Assumptions

Given: The indicator function just defined and the Euclidean coordinates $(y, s)$ on its epigraph.

[F1]

Proper, convex, and lower-semicontinuous extended-real functions on Euclidean space: properness, convexity, lower semicontinuity, and the epigraph use the stated extended-real conventions.

[F2]

Fenchel conjugate, biconjugate, and subdifferential: $f^{*} (p) = \sup_{y \in dom f} (p y - f (y))$ and $f^{* *} (x)$ is the supremum over $p$ with finite $f^{*} (p)$ .

[F3]

Extended-real convexity and lower semicontinuity are characterized by the epigraph: convexity and lower semicontinuity are equivalent to convexity and closedness of the epigraph.

[L1]

Strictly separate a point from a nonempty closed convex Euclidean set: a point outside a nonempty closed convex Euclidean set has a strict linear separator.

[L2]

Finite-dimensional Fenchel-Moreau biconjugation: a proper closed convex extended-real function on finite-dimensional Euclidean space equals its biconjugate.

Verification

technique · direct calculation

1.1F1F3L1givenalgebra

The domain is $[0, \infty)$ , so it is nonempty and $f$ is proper. Its epigraph is $E = {(y, s) : y \geq 0, s \geq 0}$ , a nonempty closed convex set; [F3] therefore makes $f$ closed and convex. The point $(- 1, 0)$ lies outside $E$ . With $u = (- 1, 0)$ and $c = 0$ , every $(y, s) \in E$ satisfies $⟨ u, (y, s) ⟩ = - y \leq 0 < 1 = ⟨ u, (- 1, 0) ⟩$ , so this is the strict vertical separator; its height coefficient is zero.

1.2F2givenalgebrachoose

For $p \leq 0$ and $y \geq 0$ , $p y \leq 0$ , with equality at $y = 0$ , so $f^{*} (p) = 0$ . For $p > 0$ , given any real $M$ choose $y > \max {0, M / p}$ ; then $p y > M$ , so the defining supremum is $+ \infty$ . Therefore $dom f^{*} = (- \infty, 0]$ .

2.1F2step 1.2algebrachoose

At $x = - 1$ , the biconjugate is $\sup_{p \leq 0} (- p - f^{*} (p)) = \sup_{p \leq 0} (- p) = + \infty$ : for every real bound $M$ , the admissible choice $p = - \max {M, 0} - 1$ gives $- p > M$ . Thus $f^{* *} (- 1) = f (- 1)$ .

3.1F2L2step 1.2step 2.1algebra∎

For each $k \geq 0$ and $y \geq 0$ , $ℓ_{k} (y) = - k y \leq 0 = f (y)$ , so $ℓ_{k}$ is an affine minorant. Its value at $- 1$ is $k$ , and the conjugate choice $p = - k$ has $f^{*} (- k) = 0$ , so the same family realizes arbitrarily large dual values at the point outside the domain. The conclusion is consistent with the full equality in [L2].

Depends on

Dependency tree · 2 levels

Sources

R. T. Rockafellar, Convex Analysis, §12, Theorems 12.1–12.2