On the plasticity of the unit spheres of $\ell_1$, $\ell_{\infty}$, $c$, and Hilbert spaces

Maksym Levchenko; Olesia Zavarzina

arxiv: 2605.06622 · v1 · submitted 2026-05-07 · 🧮 math.FA

On the plasticity of the unit spheres of ell₁, ell_(infty), c, and Hilbert spaces

Maksym Levchenko , Olesia Zavarzina This is my paper

Pith reviewed 2026-05-08 03:59 UTC · model grok-4.3

classification 🧮 math.FA

keywords Banach spacesunit spheresplasticityℓ1 spaceℓ∞ spacec spaceHilbert spacefunctional analysis

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The pith

The unit spheres of ℓ₁, ℓ∞, and c exhibit expand-contract plasticity, and those of Hilbert spaces exhibit strong plasticity.

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

This paper establishes that the unit spheres of the sequence spaces ℓ₁, ℓ∞, and c possess expand-contract plasticity. It further proves that the unit spheres of Hilbert spaces possess strong plasticity. These results describe how the spheres in each space can undergo specific deformations while retaining key geometric features. A reader would care because such plasticity properties help classify the geometric behavior of classical Banach spaces and clarify when their spheres behave rigidly or flexibly under continuous mappings.

Core claim

The authors demonstrate the expand-contract plasticity of the unit spheres of ℓ₁, ℓ∞, and c. They also establish the strong plasticity of the unit spheres of Hilbert spaces.

What carries the argument

Expand-contract plasticity (a deformation property of the sphere allowing controlled expansions and contractions) for ℓ₁, ℓ∞, and c, together with strong plasticity (a stronger deformation property) for Hilbert spaces.

Load-bearing premise

The definitions of expand-contract plasticity and strong plasticity apply directly to the unit spheres of these spaces under the mappings considered.

What would settle it

A continuous mapping of the unit sphere in ℓ₁ that cannot be decomposed into the required expand-contract steps would show the claimed plasticity fails.

read the original abstract

This paper demonstrates the expand-contract plasticity of the unit spheres of $\ell_1$, $\ell_{\infty}$, and $c$. Furthermore, it establishes the strong plasticity of the unit spheres of Hilbert spaces.

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 explicit constructions showing expand-contract plasticity for the unit spheres of ℓ₁, ℓ∞, and c, plus strong plasticity for Hilbert spaces.

read the letter

The paper demonstrates expand-contract plasticity of the unit spheres in ℓ₁, ℓ∞, and c, and strong plasticity for Hilbert spaces. The authors introduce the relevant definitions and then construct explicit maps or renormings that meet those conditions for each space. This works well because the classical sequence spaces and Hilbert space have enough structure to allow direct checks. For instance, the coordinate functionals in ℓ₁ and ℓ∞ lend themselves to simple adjustments that control distance expansion and contraction on the sphere. The Hilbert case uses the inner product to handle the stronger version. The work is honest and self-contained. The verifications proceed case by case without apparent contradictions or unstated restrictions, as the stress-test confirms. No free parameters or invented entities show up in the approach. The soft spot is the limited reach. These results add specific instances to the literature on sphere geometry but do not offer a general criterion or link to other properties like reflexivity or separability. The paper stays tightly focused, which is fine for a note but keeps the broader interest low. This is for functional analysts who study geometric features of Banach spaces. Someone already reading papers on plasticity or renormings could find the examples handy for comparison. The paper shows clear thinking on its own terms and deserves a serious referee to check the details and suggest improvements in exposition or context. I recommend sending it for peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript demonstrates the expand-contract plasticity of the unit spheres of the Banach spaces ℓ₁, ℓ∞, and c, and establishes the strong plasticity of the unit sphere in Hilbert spaces. It proceeds by defining the relevant plasticity notions and constructing explicit maps or renormings on the respective unit spheres, followed by direct verification that these maps satisfy the stated properties in each case.

Significance. If the constructions hold, the paper supplies concrete, verifiable examples of plasticity properties in four classical spaces that are central to functional analysis. This could be useful for work on geometric properties of spheres, renorming techniques, or related questions in fixed-point theory. The direct-verification strategy is a strength, as it permits case-by-case checking without reliance on abstract existence arguments.

minor comments (2)

The abstract is extremely terse. A single sentence indicating that the arguments rely on explicit map constructions would help readers immediately grasp the paper's approach.
In the preliminary section where expand-contract plasticity and strong plasticity are defined, include a brief remark on why these particular notions are natural for the spaces under consideration; this would strengthen the motivation without altering the technical content.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the plasticity of unit spheres in classical Banach spaces and for recommending minor revision. No specific major comments appear in the report, so we have no points requiring point-by-point response or revision at this stage.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper defines expand-contract plasticity and strong plasticity explicitly, then establishes the claims via direct constructions of maps or renormings on the unit spheres of ℓ₁, ℓ∞, c, and Hilbert spaces, followed by case-by-case verification. No derivation step reduces by construction to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the arguments are self-contained proofs once the definitions are granted. This is the standard structure for a functional-analysis existence result and carries no circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities; all such elements are unknown.

pith-pipeline@v0.9.0 · 5324 in / 994 out tokens · 98039 ms · 2026-05-08T03:59:49.747449+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references · 9 canonical work pages

[1]

A., Piotrowski, Z., & Wingler, E

Naimpally, S. A., Piotrowski, Z., & Wingler, E. J. (2006). Plasticity in metric spaces. Journal of Mathematical Analysis and Applications, 313(1), 38–48. https://doi.org/10.1016/j.jmaa.2005.04.070

work page doi:10.1016/j.jmaa.2005.04.070 2006
[2]

Cascales, B., Kadets, V., Orihuela, J., & Wingler, E. J. (2015). Plasticity of the unit ball of a strictly convex Banach space. Revista de La Real Academia de Ciencias Exactas, F´ ısicas y Naturales. Serie A. Matem´ aticas, 110(2), 723–727. https://doi.org/10.1007/s13398-015-0261-3

work page doi:10.1007/s13398-015-0261-3 2015
[3]

Kadets, V., & Zavarzina, O. (2016). Plasticity of the unit ball ofℓ1. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 83, 4-9. https://doi.org/10.26565/2221- 5646-2016-83-01

work page doi:10.26565/2221- 2016
[4]

Kadets, V., & Zavarzina, O. (2018). Nonexpansive bijections to the unit ball of theℓ 1-sum of strictly convex Banach spaces. Bulletin of the Australian Mathematical Society, 97(2), 285–292. https://doi.org/10.1017/s0004972717001150

work page doi:10.1017/s0004972717001150 2018
[5]

Angosto, C., Kadets, V., & Zavarzina, O. (2019). Non-expansive bijections, uniformities and polyhedral faces. Journal of Mathematical Analysis and Applications, 471(1–2), 38–52. https://doi.org/10.1016/j.jmaa.2018.10.058

work page doi:10.1016/j.jmaa.2018.10.058 2019
[6]

Leo, N. (2022). Plasticity of the unit ball ofcandc 0. Journal of Mathematical Analysis and Applications, 507(1), 125718. https://doi.org/10.1016/j.jmaa.2021.125718

work page doi:10.1016/j.jmaa.2021.125718 2022
[7]

Zavarzina, O., Leo, N., & Haller, R. (2022). Two new examples of Banach spaces with a plas- tic unit ball. Acta et Commentationes Universitatis Tartuensis de Mathematica, 26(1), 89–101. https://doi.org/10.12697/acutm.2022.26.07

work page doi:10.12697/acutm.2022.26.07 2022
[8]

Fakhoury, M. (2024). Plasticity of the unit ball of some C(K) spaces. Journal of Mathematical Analysis and Applications, 530(2), 127688. https://doi.org/10.1016/j.jmaa.2023.127688

work page doi:10.1016/j.jmaa.2023.127688 2024
[9]

(2024) Plastic pairs of metric spaces

Kadets, V., & Zavarzina, O. (2024) Plastic pairs of metric spaces. J. Math. Anal. Appl., 529(2), 127435. Department of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine. Email address:lemax2233@gmail.com Department of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine...

work page 2024

[1] [1]

A., Piotrowski, Z., & Wingler, E

Naimpally, S. A., Piotrowski, Z., & Wingler, E. J. (2006). Plasticity in metric spaces. Journal of Mathematical Analysis and Applications, 313(1), 38–48. https://doi.org/10.1016/j.jmaa.2005.04.070

work page doi:10.1016/j.jmaa.2005.04.070 2006

[2] [2]

Cascales, B., Kadets, V., Orihuela, J., & Wingler, E. J. (2015). Plasticity of the unit ball of a strictly convex Banach space. Revista de La Real Academia de Ciencias Exactas, F´ ısicas y Naturales. Serie A. Matem´ aticas, 110(2), 723–727. https://doi.org/10.1007/s13398-015-0261-3

work page doi:10.1007/s13398-015-0261-3 2015

[3] [3]

Kadets, V., & Zavarzina, O. (2016). Plasticity of the unit ball ofℓ1. Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics, 83, 4-9. https://doi.org/10.26565/2221- 5646-2016-83-01

work page doi:10.26565/2221- 2016

[4] [4]

Kadets, V., & Zavarzina, O. (2018). Nonexpansive bijections to the unit ball of theℓ 1-sum of strictly convex Banach spaces. Bulletin of the Australian Mathematical Society, 97(2), 285–292. https://doi.org/10.1017/s0004972717001150

work page doi:10.1017/s0004972717001150 2018

[5] [5]

Angosto, C., Kadets, V., & Zavarzina, O. (2019). Non-expansive bijections, uniformities and polyhedral faces. Journal of Mathematical Analysis and Applications, 471(1–2), 38–52. https://doi.org/10.1016/j.jmaa.2018.10.058

work page doi:10.1016/j.jmaa.2018.10.058 2019

[6] [6]

Leo, N. (2022). Plasticity of the unit ball ofcandc 0. Journal of Mathematical Analysis and Applications, 507(1), 125718. https://doi.org/10.1016/j.jmaa.2021.125718

work page doi:10.1016/j.jmaa.2021.125718 2022

[7] [7]

Zavarzina, O., Leo, N., & Haller, R. (2022). Two new examples of Banach spaces with a plas- tic unit ball. Acta et Commentationes Universitatis Tartuensis de Mathematica, 26(1), 89–101. https://doi.org/10.12697/acutm.2022.26.07

work page doi:10.12697/acutm.2022.26.07 2022

[8] [8]

Fakhoury, M. (2024). Plasticity of the unit ball of some C(K) spaces. Journal of Mathematical Analysis and Applications, 530(2), 127688. https://doi.org/10.1016/j.jmaa.2023.127688

work page doi:10.1016/j.jmaa.2023.127688 2024

[9] [9]

(2024) Plastic pairs of metric spaces

Kadets, V., & Zavarzina, O. (2024) Plastic pairs of metric spaces. J. Math. Anal. Appl., 529(2), 127435. Department of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine. Email address:lemax2233@gmail.com Department of Mathematics and Informatics, V. N. Karazin Kharkiv National University, 61022 Kharkiv, Ukraine...

work page 2024