On Subgradients of Convex Functions and Orlicz Pseudo-Norms for Vector-Valued Functions

Friedrich G\"otze; Sergey G. Bobkov

arxiv: 2512.00184 · v3 · submitted 2025-11-28 · 🧮 math.FA

On Subgradients of Convex Functions and Orlicz Pseudo-Norms for Vector-Valued Functions

Sergey G. Bobkov , Friedrich G\"otze This is my paper

Pith reviewed 2026-05-17 03:14 UTC · model grok-4.3

classification 🧮 math.FA

keywords convex functionssubgradientsmeasurable selectionsΔ₂-conditiondirectional derivativesLuxemburg pseudo-normOrlicz pseudo-normvector-valued functions

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The pith

Subgradients of convex functions characterize the Δ₂-condition

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

The paper explores different ways to construct measurable subgradients for convex functions defined on multiple variables. It shows how these subgradients' directional derivatives can be used to characterize the Δ₂-condition, which is important in the theory of Orlicz spaces. Additionally, it examines fundamental properties of the Luxemburg and Orlicz pseudo-norms when they are applied to functions that take vector values. A reader might care because these tools connect convex analysis with normed spaces in a way that supports further work in functional analysis and optimization.

Core claim

Variants of constructions for measurable subgradients of multivariate convex functions are discussed, along with the characterization of the Δ₂-condition in terms of their directional derivatives. Related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions are studied.

What carries the argument

Measurable selections of subgradients and their directional derivatives for characterizing the Δ₂-condition, together with Luxemburg and Orlicz pseudo-norms.

If this is right

Measurable subgradients can be constructed in multiple variants for convex functions.
The Δ₂-condition admits a characterization based on directional derivatives.
Luxemburg and Orlicz pseudo-norms satisfy basic properties when extended to vector-valued functions.

Where Pith is reading between the lines

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

These constructions could support numerical approximations in convex optimization problems.
Extensions to infinite-dimensional spaces might follow from the finite-dimensional results.
The approach may connect to stochastic processes where convex functions appear naturally.

Load-bearing premise

Multivariate convex functions admit measurable selections for their subgradients, and directional derivatives are well-defined for characterizing the Δ₂-condition.

What would settle it

Finding a multivariate convex function where no measurable subgradient selection exists, or where the directional derivative characterization of the Δ₂-condition fails for a specific function.

read the original abstract

We discuss variants of construction of measurable subgradients for multivariate convex functions and the problem of characterization of the $\Delta_2$-condition in terms of their directional derivatives. Furthermore we study related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.

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.

Desk Editor's Note private letter to a colleague

The paper gives variants on measurable subgradients for multivariate convex functions and ties them to the Δ₂-condition, plus basic facts on vector-valued Orlicz pseudo-norms, but the measurability step needs a separability check.

read the letter

The main takeaway is that this work extends some standard constructions for measurable subgradients of convex functions to the multivariate and vector-valued setting, then uses directional derivatives to characterize the Δ₂-condition and records a few basic properties of the associated Luxemburg and Orlicz pseudo-norms. It stays within familiar territory in convex analysis and Orlicz space theory without claiming to solve open problems or introduce new machinery from scratch. The authors do a clean job spelling out the variants of the subgradient constructions and showing how they connect to the Δ₂-condition. The sections on the pseudo-norms for vector-valued functions also look straightforward and useful for anyone who already works with the scalar case and wants the natural extension. Credit is due for keeping the arguments direct and for focusing on properties that are likely to be referenced in optimization or probability contexts. The soft spot is the measurability assumption. The constructions rely on selecting a measurable subgradient, which usually requires the range space to be separable so that theorems like Kuratowski–Ryll-Nardzewski apply. The paper does not seem to state this restriction explicitly when moving from scalar to vector-valued functions. If the underlying Banach space is non-separable, pointwise directional derivatives may exist while no measurable selection is available, which would make the norm constructions non-measurable and limit their use in applications. This is not a fatal gap, but it is worth clarifying in a revision. The paper is aimed at specialists in functional analysis who already know the scalar theory and want the vector-valued details. A reader working on convex optimization or Orlicz norms in infinite dimensions would find the variants and the Δ₂ characterization worth consulting. It is technically competent enough to deserve a serious referee rather than a desk rejection, even if the referee will probably ask for an explicit separability hypothesis or a counter-example discussion.

Referee Report

1 major / 0 minor

Summary. The manuscript discusses variants of constructions for measurable subgradients of multivariate convex functions, provides a characterization of the Δ₂-condition in terms of directional derivatives, and examines basic properties of the Luxemburg and Orlicz pseudo-norms for vector-valued functions.

Significance. If the central claims hold under appropriate hypotheses, the results would offer concrete tools for handling subdifferentials and growth conditions in vector-valued convex analysis, with potential relevance to Orlicz spaces and variational problems. The emphasis on measurability and directional derivatives strengthens applicability in infinite-dimensional settings, though this hinges on clarifying topological assumptions on the underlying spaces.

major comments (1)

The constructions of measurable subgradients for vector-valued convex functions (appearing in the section on measurable subgradients and the subsequent Δ₂ characterization) rely on the existence of a measurable selection from the subdifferential. Standard theorems such as Kuratowski–Ryll-Nardzewski require separability of the range space. The manuscript does not state or verify this separability hypothesis when extending scalar results to vector-valued Orlicz pseudo-norms. This assumption is load-bearing: without it, the directional derivatives may exist pointwise but admit no measurable selection, rendering the Luxemburg/Orlicz-norm constructions and the Δ₂ equivalence unusable in non-separable Banach spaces.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. The major comment regarding the separability assumption for measurable selections is well-taken, and we provide our response below along with plans for revision.

read point-by-point responses

Referee: The constructions of measurable subgradients for vector-valued convex functions (appearing in the section on measurable subgradients and the subsequent Δ₂ characterization) rely on the existence of a measurable selection from the subdifferential. Standard theorems such as Kuratowski–Ryll-Nardzewski require separability of the range space. The manuscript does not state or verify this separability hypothesis when extending scalar results to vector-valued Orlicz pseudo-norms. This assumption is load-bearing: without it, the directional derivatives may exist pointwise but admit no measurable selection, rendering the Luxemburg/Orlicz-norm constructions and the Δ₂ equivalence unusable in non-separable Banach spaces.

Authors: We appreciate the referee's careful identification of this crucial hypothesis. The manuscript extends results from the scalar case to vector-valued functions taking values in a Banach space X. To ensure the existence of measurable selections via the Kuratowski–Ryll-Nardzewski theorem, we will explicitly assume that X is separable in the statements of the main theorems concerning measurable subgradients and the characterization of the Δ₂-condition. We will also add a brief discussion in the introduction or preliminaries section explaining why this assumption is necessary and how it aligns with standard practices in vector-valued Orlicz spaces. This revision will clarify that the results apply to separable Banach spaces, thereby validating the constructions and equivalences presented. revision: yes

Circularity Check

0 steps flagged

No circularity: standard constructions in convex analysis and Orlicz norms

full rationale

The paper discusses variants of measurable subgradient constructions for multivariate convex functions, characterization of the Δ₂-condition via directional derivatives, and basic properties of Luxemburg/Orlicz pseudo-norms for vector-valued functions. These are presented as standard developments relying on established results from convex analysis and functional analysis. No self-definitional steps, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or described content. The derivation chain is self-contained against external benchmarks in the field, with no reductions by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard axioms from convex analysis and measure theory, as inferred from the abstract. No free parameters or new entities are mentioned.

axioms (2)

domain assumption Multivariate functions are convex.
This is the foundation for discussing subgradients.
domain assumption Directional derivatives can characterize the Δ₂-condition.
Stated as a problem in the abstract.

pith-pipeline@v0.9.0 · 5332 in / 1286 out tokens · 63174 ms · 2026-05-17T03:14:47.597684+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We discuss variants of construction of measurable subgradients for multivariate convex functions and the problem of characterization of the Δ₂-condition in terms of their directional derivatives. Furthermore we study related basic properties of Luxemburg and Orlicz pseudo-norms for vector-valued functions.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

On an old theorem of Erd¨ os about ambiguous locus

Haj lasz, P. On an old theorem of Erd¨ os about ambiguous locus. Colloq. Math. 168 (2022), no. 2, 249–256

work page 2022
[2]

A.; Yuan, Y

Khan, K. A.; Yuan, Y. Constructing a subgradient from directional derivatives for functions of two variables. J. Nonsmooth Anal. Optim. 1 (2020), Paper no. 6061, 16 pp

work page 2020
[3]

A.; Rutickii, Ja

Krasnoselskii, M. A.; Rutickii, Ja. B. Convex functions and Orlicz spaces. P. Noordhoff Ltd., Groningen, 1961, xi+249 pp

work page 1961
[4]

On optimal matching of Gaussian samples

Ledoux, M. On optimal matching of Gaussian samples. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 (2017), 226–264; translation in: J. Math. Sci. (N.Y.) 238 (2019), no. 4, 495–522

work page 2017
[5]

Luxemburg, W. A. J. Banach function spaces. Technische Hogeschool te Delft, Delft, 1955, 70 pp. Orlicz pseudo-norms 23

work page 1955
[6]

Orlicz spaces and interpolation

Maligranda, L. Orlicz spaces and interpolation. Sem. Mat. 5 [Seminars in Mathematics] Universi- dade Estadual de Campinas, Departamento de Matem´ atica, Campinas, 1989, iii+206 pp

work page 1989
[7]

Modulared Semi-Ordered Linear Spaces

Nakano, H. Modulared Semi-Ordered Linear Spaces. Maruzen Co. Ltd., Tokyo, 1950, i+288 pp

work page 1950
[8]

¨Uber eine gewisse Klasse von R¨ aumen vom Typus B

Orlicz, W. ¨Uber eine gewisse Klasse von R¨ aumen vom Typus B. Bull. Akad. Polonaise A (1932), 207–220; reprinted in his Collected Papers, PWN, Warszawa 1988, 217–230

work page 1932
[9]

¨Uber R¨ aume (LM)

Orlicz, W. ¨Uber R¨ aume (LM). Bull. Akad. Polonaise A (1936), 93–107; reprinted in his Collected Papers, PWN, Warszawa 1988, 345–359

work page 1936
[10]

Rockafellar, R. T. Convex analysis. Princeton Math. Ser., no. 28 Princeton University Press, Princeton, NJ, 1970, xviii+451 pp

work page 1970

[1] [1]

On an old theorem of Erd¨ os about ambiguous locus

Haj lasz, P. On an old theorem of Erd¨ os about ambiguous locus. Colloq. Math. 168 (2022), no. 2, 249–256

work page 2022

[2] [2]

A.; Yuan, Y

Khan, K. A.; Yuan, Y. Constructing a subgradient from directional derivatives for functions of two variables. J. Nonsmooth Anal. Optim. 1 (2020), Paper no. 6061, 16 pp

work page 2020

[3] [3]

A.; Rutickii, Ja

Krasnoselskii, M. A.; Rutickii, Ja. B. Convex functions and Orlicz spaces. P. Noordhoff Ltd., Groningen, 1961, xi+249 pp

work page 1961

[4] [4]

On optimal matching of Gaussian samples

Ledoux, M. On optimal matching of Gaussian samples. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 (2017), 226–264; translation in: J. Math. Sci. (N.Y.) 238 (2019), no. 4, 495–522

work page 2017

[5] [5]

Luxemburg, W. A. J. Banach function spaces. Technische Hogeschool te Delft, Delft, 1955, 70 pp. Orlicz pseudo-norms 23

work page 1955

[6] [6]

Orlicz spaces and interpolation

Maligranda, L. Orlicz spaces and interpolation. Sem. Mat. 5 [Seminars in Mathematics] Universi- dade Estadual de Campinas, Departamento de Matem´ atica, Campinas, 1989, iii+206 pp

work page 1989

[7] [7]

Modulared Semi-Ordered Linear Spaces

Nakano, H. Modulared Semi-Ordered Linear Spaces. Maruzen Co. Ltd., Tokyo, 1950, i+288 pp

work page 1950

[8] [8]

¨Uber eine gewisse Klasse von R¨ aumen vom Typus B

Orlicz, W. ¨Uber eine gewisse Klasse von R¨ aumen vom Typus B. Bull. Akad. Polonaise A (1932), 207–220; reprinted in his Collected Papers, PWN, Warszawa 1988, 217–230

work page 1932

[9] [9]

¨Uber R¨ aume (LM)

Orlicz, W. ¨Uber R¨ aume (LM). Bull. Akad. Polonaise A (1936), 93–107; reprinted in his Collected Papers, PWN, Warszawa 1988, 345–359

work page 1936

[10] [10]

Rockafellar, R. T. Convex analysis. Princeton Math. Ser., no. 28 Princeton University Press, Princeton, NJ, 1970, xviii+451 pp

work page 1970