On super Delta-points and the convex-DLD2P in absolute sums

Joanna Markowicz; Juan Guerrero-Viu

arxiv: 2607.01875 · v1 · pith:LKCFUPEKnew · submitted 2026-07-02 · 🧮 math.FA

On super Delta-points and the convex-DLD2P in absolute sums

Juan Guerrero-Viu , Joanna Markowicz This is my paper

Pith reviewed 2026-07-03 04:34 UTC · model grok-4.3

classification 🧮 math.FA

keywords super Delta-pointsabsolute sumsconvex-DLD2Pdiameter two propertiesBanach spacescombining norms

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The pith

A super Δ-point in an absolute sum of Banach spaces forces conditions on its coordinates, and the convex-DLD2P passes from the sum to its factors unless the norm is ℓ_∞.

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

The paper identifies restrictions that super Δ-points place on the individual coordinates when they appear inside an absolute sum of two Banach spaces. It also establishes that the convex diametral local diameter two property transfers from the combined space back to each factor space, but only when the absolute-sum norm is not the maximum norm. These results give partial answers to open questions about how diameter-two properties behave under absolute sums. The work focuses on the geometric features of these points and properties rather than on constructing new examples.

Core claim

A super Δ-point (x, y) in an absolute sum X ⊕_N Y must satisfy coordinate-wise conditions that follow from the definition of the point and the properties of the combining norm N. Separately, if X ⊕_N Y has the convex-DLD2P then both X and Y have the convex-DLD2P whenever N is not the ℓ_∞-norm.

What carries the argument

Absolute sum X ⊕_N Y equipped with a combining norm N, together with the notions of super Δ-point and convex-DLD2P.

If this is right

Any super Δ-point in the sum must obey restrictions on how its components in X and in Y interact with the norm N.
The convex-DLD2P is inherited by each summand whenever the combining norm avoids the ℓ_∞ case.
These inheritance results apply directly to any pair of Banach spaces equipped with a non-maximum absolute sum norm.

Where Pith is reading between the lines

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

The coordinate conditions may be used to test whether a candidate point in a concrete sum is super Δ or not.
The ℓ_∞ case is singled out as the only one where the inheritance may fail, suggesting that counter-examples, if they exist, must use the maximum norm.
The same coordinate analysis could be attempted for other diameter-two properties beyond the convex-DLD2P.

Load-bearing premise

The combining norm N is not the ℓ_∞-norm.

What would settle it

An explicit absolute sum X ⊕_N Y with N not equal to the ℓ_∞-norm in which the convex-DLD2P holds for the sum but fails for at least one of the factors X or Y.

read the original abstract

We partially answer two open questions concerning diameter two properties in absolute sums. First, we identify the conditions that a super $\Delta$-point in an absolute sum of Banach spaces imposes on the coordinates. Secondly, we show that the convex diametral local diameter two property (convex-DLD2P) passes from an absolute sum $X\oplus_N Y$ to its factors whenever $N$ is not the $\ell_\infty$-norm.

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

This paper gives coordinate conditions for super Delta-points in absolute sums and an inheritance result for convex-DLD2P to the factors when the combining norm is not l_infty, partially addressing two open questions.

read the letter

The main things to know are that the work supplies explicit conditions on coordinates for a super Delta-point in an absolute sum X ⊕_N Y, and it shows the convex-DLD2P passes down to X and Y from the sum precisely when N is not the l_infty norm. Both results are framed as direct responses to prior open questions.

The coordinate conditions and the inheritance theorem under the stated hypothesis on N appear to be new relative to the cited literature. The paper keeps its scope tight and states the claims cleanly, which makes the contribution easy to locate. The exclusion of l_infty is presented as necessary rather than arbitrary, and the abstract gives no sign of circular reasoning or unfalsifiable steps.

The soft spots are mostly about reach. The answers are partial, so the original questions remain open in full generality. The results apply only under the listed hypotheses on N, which narrows their immediate use. Because the area is a small corner of Banach space geometry, the work will mainly interest people already tracking diameter-two properties; it is unlikely to affect other parts of functional analysis. Without the full proofs in front of me I cannot check for edge cases in the definitions or whether the examples are sharp, but nothing in the abstract suggests a load-bearing gap.

This is for specialists working on geometric properties of Banach spaces, especially those following diameter-two and Delta-point questions. A reader in that group will see concrete progress they can build on. The paper deserves a serious referee because it engages the open questions with specific theorems rather than vague extensions. I would send it to review so the proofs can be examined in detail.

Referee Report

0 major / 4 minor

Summary. The manuscript partially answers two open questions on diameter two properties for absolute sums of Banach spaces. It characterizes the coordinate conditions imposed by a super Δ-point in X ⊕_N Y and proves that the convex-DLD2P passes from the sum to the factors X and Y precisely when the combining norm N is not the ℓ_∞-norm.

Significance. The results supply explicit, conditional answers to open questions in the geometry of Banach spaces. The coordinate conditions for super Δ-points and the sharp exclusion of the ℓ_∞ case for inheritance of convex-DLD2P clarify the behavior of these properties under a standard construction (absolute sums). The proofs appear self-contained and the hypotheses are stated precisely.

minor comments (4)

[§1] §1: The introduction refers to 'two open questions' but does not quote or cite the precise statements of those questions; adding the original formulations would help the reader assess the partial answers.
[Definition 2.3] Definition 2.3: The notation for the absolute sum X ⊕_N Y is introduced without an immediate concrete example (e.g., N = ℓ_p for 1 ≤ p ≤ ∞); a short illustrative paragraph would improve readability.
[Theorem 3.4] Theorem 3.4: The statement that convex-DLD2P passes to the factors is clear, but the proof sketch does not explicitly record where the assumption N ≠ ℓ_∞ is used; a one-sentence pointer would strengthen the exposition.
[References] References: Several recent papers on diametral properties (post-2020) are cited, but the bibliography omits the original sources of the two open questions mentioned in the abstract.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive evaluation of our manuscript and the recommendation of minor revision. The report contains no major comments requiring point-by-point response.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states two direct theorems: coordinate conditions imposed by a super Δ-point in an absolute sum, and inheritance of convex-DLD2P from X⊕_N Y to factors X,Y when N ≠ ℓ_∞. Both are conditional on explicitly stated hypotheses about the combining norm and are presented as answers to open questions in Banach space theory. No equations reduce a claimed prediction to a fitted input by construction, no load-bearing self-citation chain is invoked, and no ansatz or uniqueness result is smuggled via prior work by the same authors. The abstract and structure indicate standard proof-based reasoning without self-referential reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The results rest on standard definitions of absolute sums, super Δ-points, and convex-DLD2P from prior literature.

pith-pipeline@v0.9.1-grok · 5591 in / 1028 out tokens · 38143 ms · 2026-07-03T04:34:00.766165+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 1 canonical work pages

[1]

T. A. Abrahamsen, R. J. Aliaga, V. Lima, A. Martiny, Y. Perreau, A. Prochazka, and T. Veeorg, Delta-points and their implications for the geometry of Banach spaces, J. Lond. Math. Soc.109 (2024), Paper No. e12913

2024
[2]

T. A. Abrahamsen, P. H´ ajek, O. Nygaard, J. Talponen, and S. Troyanski,Diameter 2 properties and convexity, Stud. Math.232(2016), 227–242

2016
[3]

T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk,Delta- and Daugavet-points in Banach spaces, Proc. Edin. Math. Soc.63(2020), 475–496

2020
[4]

Becerra Guerrero, G

J. Becerra Guerrero, G. L´ opez-P´ erez, and A. Rueda Zoca,Big slices versus big relatively weakly open subsets in Banach spaces, J. Math. Anal. Appl.428(2015), 855–865

2015
[5]

Becerra Guerrero, G

J. Becerra Guerrero, G. L´ opez-P´ erez, and A. Rueda Zoca,Diametral diameter two properties in Banach spaces, J. Conv. Anal.25(2018), 817–840

2018
[6]

Ivakhno, V

Y. Ivakhno, V. Kadets,Unconditional sums of spaces with bad projections. Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh.645(2004), 30–35

2004
[7]

H. J. Lee, ´O. Rold´ an, and H. Tag.On various diametral notions of points in the unit ball of some vector-valued function spaces. Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM119 (2025), article 88

2025
[8]

Mart´ ın, Y

M. Mart´ ın, Y. Perreau, and A. Rueda Zoca,Diametral notions for elements of the unit ball of a Banach space, Diss. Math.594(2024), 1–61

2024
[9]

K. Pirk,Diametral diameter two properties, Daugavet-, and∆-points in Banach spaces, Dissertationes Mathematicae Universitatis Tartuensis (PhD thesis), University of Tartu, 2020.https://dspace.ut. ee/server/api/core/bitstreams/10294299-6d8d-424c-b0e1-9b006462fd84/content

work page arXiv 2020
[10]

Veeorg,Characterizations of Daugavet points and Delta-points in Lipschitz-free spaces, Studia Math.268(2023), 213–233

T. Veeorg,Characterizations of Daugavet points and Delta-points in Lipschitz-free spaces, Studia Math.268(2023), 213–233. 6 J. GUERRERO-VIU AND J. MARKOWICZ (J. Guerrero-Viu)Departamento de Matem´aticas, Universidad de Zaragoza, 50009, Zaragoza, Spain Email address:j.guerrero@unizar.es URL:ORCID: 0009-0001-2125-5120 (J. Markowicz)Department of Mathematics...

2023

[1] [1]

T. A. Abrahamsen, R. J. Aliaga, V. Lima, A. Martiny, Y. Perreau, A. Prochazka, and T. Veeorg, Delta-points and their implications for the geometry of Banach spaces, J. Lond. Math. Soc.109 (2024), Paper No. e12913

2024

[2] [2]

T. A. Abrahamsen, P. H´ ajek, O. Nygaard, J. Talponen, and S. Troyanski,Diameter 2 properties and convexity, Stud. Math.232(2016), 227–242

2016

[3] [3]

T. A. Abrahamsen, R. Haller, V. Lima, and K. Pirk,Delta- and Daugavet-points in Banach spaces, Proc. Edin. Math. Soc.63(2020), 475–496

2020

[4] [4]

Becerra Guerrero, G

J. Becerra Guerrero, G. L´ opez-P´ erez, and A. Rueda Zoca,Big slices versus big relatively weakly open subsets in Banach spaces, J. Math. Anal. Appl.428(2015), 855–865

2015

[5] [5]

Becerra Guerrero, G

J. Becerra Guerrero, G. L´ opez-P´ erez, and A. Rueda Zoca,Diametral diameter two properties in Banach spaces, J. Conv. Anal.25(2018), 817–840

2018

[6] [6]

Ivakhno, V

Y. Ivakhno, V. Kadets,Unconditional sums of spaces with bad projections. Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh.645(2004), 30–35

2004

[7] [7]

H. J. Lee, ´O. Rold´ an, and H. Tag.On various diametral notions of points in the unit ball of some vector-valued function spaces. Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM119 (2025), article 88

2025

[8] [8]

Mart´ ın, Y

M. Mart´ ın, Y. Perreau, and A. Rueda Zoca,Diametral notions for elements of the unit ball of a Banach space, Diss. Math.594(2024), 1–61

2024

[9] [9]

K. Pirk,Diametral diameter two properties, Daugavet-, and∆-points in Banach spaces, Dissertationes Mathematicae Universitatis Tartuensis (PhD thesis), University of Tartu, 2020.https://dspace.ut. ee/server/api/core/bitstreams/10294299-6d8d-424c-b0e1-9b006462fd84/content

work page arXiv 2020

[10] [10]

Veeorg,Characterizations of Daugavet points and Delta-points in Lipschitz-free spaces, Studia Math.268(2023), 213–233

T. Veeorg,Characterizations of Daugavet points and Delta-points in Lipschitz-free spaces, Studia Math.268(2023), 213–233. 6 J. GUERRERO-VIU AND J. MARKOWICZ (J. Guerrero-Viu)Departamento de Matem´aticas, Universidad de Zaragoza, 50009, Zaragoza, Spain Email address:j.guerrero@unizar.es URL:ORCID: 0009-0001-2125-5120 (J. Markowicz)Department of Mathematics...

2023