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arxiv: 2504.07631 · v2 · submitted 2025-04-10 · 🧮 math.FA

The super Alternative Daugavet property for Banach spaces

Johann Langemets , Marcus L\~oo , Miguel Mart\'in , Yo\"el Perreau , Abraham Rueda Zoca This is my paper

Pith reviewed 2026-05-22 20:50 UTC · model grok-4.3

classification 🧮 math.FA
keywords super alternative Daugavet propertyDaugavet propertyalternative Daugavet propertyrough normpoint of continuity propertyRadon-Nikodym propertyBanach spacesvector-valued functions
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The pith

The super alternative Daugavet property sits strictly between the Daugavet property and the alternative Daugavet property in Banach spaces.

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

The paper defines the super alternative Daugavet property for a Banach space as the requirement that for any unit sphere element x and any relatively weakly open subset W of the unit ball that meets the sphere, there is a y in W and a unit scalar theta making the norm of x plus theta y nearly equal to two. This property is possessed by all spaces with the Daugavet property and in turn implies the alternative Daugavet property. The authors construct examples separating super ADP from the Daugavet property on one side and from the alternative version on the other. They prove that super ADP forces the norm to be rough, preventing the space from being Asplund, and causes failure of the point of continuity property, which rules out the Radon-Nikodym property. They also examine localized versions of the three properties and derive characterizations for spaces of vector-valued continuous and integrable functions.

Core claim

A Banach space X has the super ADP if for every x in the unit sphere and every relatively weakly open W of the unit ball intersecting the sphere, one can find y in W and |theta|=1 such that ||x + theta y|| is almost two. This condition is known to hold in Daugavet spaces and to imply the alternative Daugavet property. The paper shows strict separation with examples, proves that the norm must be rough so X is not Asplund and fails the point of continuity property in particular the Radon-Nikodym property, and provides characterizations in vector-valued function spaces.

What carries the argument

The super alternative Daugavet property, a condition on norms of sums with phased vectors from weakly open sets that strengthens the alternative Daugavet property while being weaker than the full Daugavet property.

Load-bearing premise

The definition and strict inclusions rely on the existence of relatively weakly open subsets of the unit ball that intersect the unit sphere in a way that allows the norm condition to be satisfied independently of the full Daugavet property.

What would settle it

An Asplund Banach space that satisfies the super ADP definition would falsify the claim that the property forces a rough norm and excludes Asplund spaces.

Figures

Figures reproduced from arXiv: 2504.07631 by Abraham Rueda Zoca, Johann Langemets, Marcus L\~oo, Miguel Mart\'in, Yo\"el Perreau.

Figure 1
Figure 1. Figure 1: Relations between the notions Our next aim is to illustrate the difference between super AD and super Daugavet points, showing that the later are super AD for all directions at the same time. Proposition 4.7. Let x ∈ SX be a super Daugavet point. Then, for every ε > 0 and every non-empty relatively weakly open subset W of BX , there exists y ∈ W such that kx + θyk > 2−ε for every θ ∈ T. Proof. From the def… view at source ↗
read the original abstract

We introduce the super alternative Daugavet property (super ADP) which lies strictly between the Daugavet property and the Alternative Daugavet property as follows. A Banach space $X$ has the super ADP if for every element $x$ in the unit sphere and for every relatively weakly open subset $W$ of the unit ball intersecting the unit sphere, one can find an element $y\in W$ and a modulus one scalar $\theta$ such that $\|x+\theta y\|$ is almost two. It is known that spaces with the Daugavet property satisfy this condition, and that this condition implies the Alternative Daugavet property. We first provide examples of super ADP spaces which fail the Daugavet property. We show that the norm of a super ADP space is rough, hence the space cannot be Asplund, and we also prove that the space fails the point of continuity property (particularly, the Radon--Nikod\'ym property). In particular, we get examples of spaces with the Alternative Daugavet property that fail the super ADP. For a better understanding of the differences between the super ADP, the Daugavet property, and the Alternative Daugavet property, we will also consider the localizations of these three properties and prove that they behave rather differently. As a consequence, we provide characterizations of the super ADP for spaces of vector-valued continuous functions and of vector-valued integrable 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.

Referee Report

2 major / 3 minor

Summary. The paper introduces the super alternative Daugavet property (super ADP) for a Banach space X: for every x in the unit sphere and every relatively weakly open W subset of the unit ball with W intersecting the sphere, there exist y in W and |θ|=1 such that ||x + θ y|| is arbitrarily close to 2. It establishes the chain DP ⇒ super ADP ⇒ ADP with strict inclusions, supplies explicit examples separating the three properties, proves that super ADP implies a rough norm (hence X is not Asplund) and fails both the point-of-continuity property and the Radon-Nikodým property, examines the local versions of the three properties, and derives characterizations of super ADP for C(K,X) and L¹(μ,X).

Significance. If the results hold, the work inserts a new intermediate property in the Daugavet hierarchy, supplies concrete separating examples, and gives useful characterizations for vector-valued function spaces. The analysis of local versions and the geometric consequences (roughness, failure of RNP/PCP) are substantive contributions to the study of norm geometry in Banach spaces.

major comments (2)
  1. [§3] §3 (examples separating super ADP from DP): the verification that the constructed spaces satisfy the super ADP condition for every relatively weakly open W intersecting the sphere is only outlined; an explicit check that the chosen y and θ work uniformly for all such W is needed to confirm the strict separation from DP.
  2. [§5] Theorem on characterizations for C(K,X) (likely §5): the statement reduces super ADP of C(K,X) to a condition on X, but the proof sketch does not address the case when K is not metrizable; the reduction step should be verified for general compact K to ensure the characterization is load-bearing.
minor comments (3)
  1. [Definition 1.1] Notation: the phrase 'almost two' is used repeatedly without a uniform ε-quantifier; replace with an explicit 'for every ε>0 there exist...' formulation for clarity.
  2. [§4] The local versions of DP, super ADP, and ADP are introduced in §4 but their mutual relations are stated without a diagram or table; a small comparison table would improve readability.
  3. [Introduction] Missing reference: the claim that ADP spaces need not be Asplund should cite the relevant prior work on ADP (e.g., the original ADP paper) rather than only the current manuscript.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (examples separating super ADP from DP): the verification that the constructed spaces satisfy the super ADP condition for every relatively weakly open W intersecting the sphere is only outlined; an explicit check that the chosen y and θ work uniformly for all such W is needed to confirm the strict separation from DP.

    Authors: We agree with the referee that the verification in §3 is outlined rather than fully explicit. In the revised version of the manuscript, we will include a detailed explicit check demonstrating that the chosen y and θ work for all such relatively weakly open sets W, uniformly confirming the super ADP condition and the strict separation from the Daugavet property. revision: yes

  2. Referee: [§5] Theorem on characterizations for C(K,X) (likely §5): the statement reduces super ADP of C(K,X) to a condition on X, but the proof sketch does not address the case when K is not metrizable; the reduction step should be verified for general compact K to ensure the characterization is load-bearing.

    Authors: The characterization theorem for C(K,X) is intended to hold for general compact Hausdorff spaces K. While the proof sketch emphasizes the metrizable case for simplicity, the underlying arguments extend to the non-metrizable setting. We will revise the proof to explicitly address and verify the reduction step for arbitrary compact K, making the characterization fully load-bearing. revision: yes

Circularity Check

0 steps flagged

No significant circularity: new property defined directly from norm and weak topology with independent examples and proofs

full rationale

The paper introduces super ADP via an explicit definition using the unit sphere, relatively weakly open subsets of the unit ball, and the norm condition ||x + θ y|| ≈ 2. It states the chain DP ⇒ super ADP ⇒ ADP as previously known without deriving them from the new definition. Strict separation is established by explicit examples of spaces satisfying super ADP but not DP, and ADP but not super ADP. Further results on norm roughness, failure of PCP/RNP, localizations, and characterizations for C(K,X) and L¹(μ,X) are derived from the definition and standard Banach space techniques without self-referential equations, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the claims. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the standard axioms of Banach space theory and the weak topology; the new property itself is the main addition with no free parameters or external entities postulated.

axioms (1)
  • standard math Standard properties of norms, unit balls, and relatively weakly open sets in Banach spaces
    Invoked throughout the definition of super ADP and all implications.
invented entities (1)
  • super alternative Daugavet property no independent evidence
    purpose: Intermediate condition between Daugavet property and alternative Daugavet property
    Newly defined property; no independent falsifiable evidence outside the paper is provided in the abstract.

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