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arxiv: 2509.21863 · v5 · pith:PBR47S6Knew · submitted 2025-09-26 · 🧮 math.FA · math.OC

Gamma-Convergence of Convex Functions, Conjugates, and Subdifferentials

Rafael Correa , Pedro P\'erez-Aros , Jos\'e Pablo Santander This is my paper

Pith reviewed 2026-05-18 13:17 UTC · model grok-4.3

classification 🧮 math.FA math.OC
keywords Gamma-convergenceconvex functionsFenchel conjugatessubdifferentialsweakly compactly generated Banach spacesequicoercivitygraphical convergenceintegral functionals
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The pith

In WCG Banach spaces, Gamma-convergence of convex functions is equivalent to Gamma-convergence of their Fenchel conjugates and to graphical convergence of their subdifferentials under equicoercivity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the duality principle for Gamma-convergence of convex lower semicontinuous functions, previously known only in separable reflexive Banach spaces, to the broader setting of weakly compactly generated Banach spaces. It proves that equicoercivity implies equivalence between Gamma-convergence in the norm topology, Gamma-convergence of the Fenchel conjugates in the weak-star topology, and graphical convergence of the subdifferentials in the product topology. This extension covers separable spaces without reflexivity, reflexive spaces without separability, and spaces such as L1, and it yields dual characterizations for Gamma-convergence of convex integral functionals on Lp spaces.

Core claim

Under the classical hypothesis of equicoercivity, Gamma-convergence in the norm topology is equivalent to Gamma-convergence of the Fenchel conjugates in the weak-star topology, and both are equivalent to graphical convergence of the associated subdifferentials with respect to the product topology given by the norm on the primal space and the weak-star topology on the dual, when the underlying space is weakly compactly generated.

What carries the argument

The duality principle that equates Gamma-convergence of a sequence of convex functions to Gamma-convergence of their Fenchel conjugates and to graphical convergence of their subdifferentials, enabled by equicoercivity in WCG spaces.

If this is right

  • The equivalence applies directly to convex integral functionals on L^p spaces for 1 ≤ p < ∞.
  • The result holds for all separable Banach spaces and all reflexive Banach spaces.
  • The duality covers important non-reflexive and non-separable spaces such as L^1(μ) for arbitrary σ-finite measures.
  • Dual characterizations of Gamma-convergence become available for a wider range of variational problems in functional analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The equivalence may simplify stability analysis for optimization problems posed directly in non-reflexive spaces.
  • Similar duality results could be investigated in spaces with weaker compactness properties if suitable uniform integrability conditions replace full WCG.
  • Graphical convergence of subdifferentials under this topology offers a potential tool for studying sensitivity of solutions in infinite-dimensional variational inequalities.

Load-bearing premise

The underlying Banach space is weakly compactly generated.

What would settle it

A sequence of equicoercive convex lower semicontinuous functions on a non-WCG Banach space that Gamma-converges in the norm topology but whose Fenchel conjugates fail to Gamma-converge in the weak-star topology.

read the original abstract

We extend the duality principle for the $\Gamma$-convergence of convex lower semicontinuous functions, which was previously established only in separable reflexive Banach spaces, to the broader class of weakly compactly generated (WCG) Banach spaces, addressing a question of Fitzpatrick and Lewis. Under the same classical hypothesis of equicoercivity, we show that $\Gamma$-convergence in the norm topology is equivalent to $\Gamma$-convergence of the Fenchel conjugates in the weak$^\ast$ topology. We further prove that this duality is equivalent to the graphical convergence of the associated subdifferentials with respect to the product topology given by the norm on the primal space and the weak$^\ast$ topology on the dual. The WCG setting encompasses all separable and all reflexive Banach spaces separately, i.e, separable spaces without reflexivity assumptions and reflexive spaces without separability assumptions, as well as important non-reflexive spaces which may fail to be separable, such as $L^1(\mu)$ for an arbitrary $\sigma$-finite measure. As an application, we derive dual characterizations of the $\Gamma$-convergence of convex integral functionals on $L^p$ spaces ($1\leq p<\infty $).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper extends the duality principle for Gamma-convergence of convex lower semicontinuous functions from separable reflexive Banach spaces to weakly compactly generated (WCG) Banach spaces. Under the equicoercivity hypothesis, it establishes that Gamma-convergence in the norm topology is equivalent to Gamma-convergence of the Fenchel conjugates in the weak* topology, and that both are equivalent to graphical convergence of the subdifferentials in the product topology (norm on the primal space and weak* on the dual). The result is applied to obtain dual characterizations of Gamma-convergence for convex integral functionals on L^p spaces (1 ≤ p < ∞).

Significance. If the proofs are correct, the extension resolves a question posed by Fitzpatrick and Lewis and meaningfully broadens the setting to include all reflexive spaces (without separability) and important non-reflexive non-separable spaces such as L^1(μ) for arbitrary σ-finite measures. The WCG framework is a natural and technically appropriate choice that preserves the equicoercivity hypothesis while enabling functional-analytic arguments. The application to integral functionals supplies a concrete illustration of utility in variational analysis. The stress-test concern about sequential versus net-based Gamma-convergence does not land: the manuscript defines Gamma-convergence via nets in the weak* topology and uses the WCG property to reduce arguments appropriately, so the claimed equivalences are not undermined by metrizability issues.

major comments (1)
  1. [§3] §3, Theorem 3.2: the graphical convergence statement for subdifferentials is load-bearing for the full equivalence chain; the proof sketch invokes a WCG-specific selection argument that should be expanded to show explicitly how the product topology (norm × weak*) interacts with the equicoercivity hypothesis without additional separability assumptions.
minor comments (3)
  1. [Abstract] Abstract: the clause 'separately, i.e, separable spaces' contains a missing comma after 'i.e.' and should read 'i.e., separable spaces'.
  2. [§2.1] §2.1: the notation for the weak* topology on X* is introduced without an explicit reminder that it is the topology of pointwise convergence on X; adding one sentence would improve readability for readers outside convex analysis.
  3. [§5] §5: the application to integral functionals on L^p would benefit from a brief remark on how the WCG property of L^p (p < ∞) is verified when the underlying measure space is non-separable.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and positive evaluation of our manuscript. We appreciate the recognition that the extension to WCG spaces addresses the question of Fitzpatrick and Lewis while preserving the equicoercivity hypothesis. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3] §3, Theorem 3.2: the graphical convergence statement for subdifferentials is load-bearing for the full equivalence chain; the proof sketch invokes a WCG-specific selection argument that should be expanded to show explicitly how the product topology (norm × weak*) interacts with the equicoercivity hypothesis without additional separability assumptions.

    Authors: We agree that a more detailed exposition of the proof of Theorem 3.2 will enhance readability and rigor. In the revised manuscript we will expand the argument by first recalling the WCG selection principle (a bounded set in the dual admits a weak*-dense countable subset whose closed convex hull is weak*-compact), then showing step-by-step how equicoercivity produces a uniform bound on the subdifferentials that allows us to extract a net in the product topology (norm on X, weak* on X*). We will explicitly verify that any weak*-limit point of the conjugates corresponds to a graphical limit point of the subdifferentials without invoking separability of X or X*, relying only on the WCG property to guarantee the necessary compactness and selection. This expansion will make the interaction between the topologies and the coercivity hypothesis fully transparent while preserving the net-based definition of Γ-convergence already used in the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: direct functional-analytic extension under equicoercivity

full rationale

The paper extends the known duality principle for Gamma-convergence of convex lsc functions from separable reflexive spaces to WCG Banach spaces by proving equivalences between norm Gamma-convergence, weak* Gamma-convergence of conjugates, and graphical subdifferential convergence. These are established directly via standard convex analysis tools and the WCG property, without any self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the central claims. The equicoercivity hypothesis is external and classical; the WCG assumption is used to handle non-separable cases but does not create a definitional loop. The derivation remains self-contained against external benchmarks in convex analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of WCG Banach spaces and the classical equicoercivity assumption already used in prior work; no new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption The underlying space is a weakly compactly generated Banach space.
    Invoked to extend the duality beyond separable reflexive spaces.
  • domain assumption Equicoercivity of the sequence of functions.
    Classical hypothesis retained from the separable-reflexive case to obtain the equivalences.

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