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arxiv: 2606.07370 · v1 · pith:NQNCNXH3new · submitted 2026-06-05 · 🧮 math.FA

On strengthened versions of Klee's convex body problem in Banach spaces

Lixin Cheng , Wuyi He , Chulei Liu , Zhizheng Yu This is my paper

Pith reviewed 2026-06-27 20:33 UTC · model grok-4.3

classification 🧮 math.FA
keywords Banach spaceconvex bodyKlee's problemuniform convexityuniform smoothnesssuperreflexivityHausdorff metricAsplund space
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The pith

Superreflexivity of a Banach space is equivalent to uniform approximation of every convex body by uniformly convex or uniformly smooth bodies.

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

The paper proves multiple equivalences linking properties of a Banach space X to how well its convex bodies can be approximated. It shows that X admits an equivalent locally uniformly convex norm if and only if every convex body can be approximated in the Hausdorff metric by locally uniformly convex bodies, and gives parallel results for Fréchet smoothness under reflexivity or the Asplund condition in the separable case. The strongest result equates superreflexivity of X with the possibility of uniform approximation of convex bodies by uniformly convex bodies, by uniformly smooth bodies, or by both. A sympathetic reader cares because these statements turn geometric approximation questions into precise characterizations of the space's renorming properties.

Core claim

The following statements are equivalent: A. X is super reflexive; B. Every convex body in X can be uniformly approximated by uniformly convex bodies; C. Every convex body in X can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in X can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies. The paper also establishes that every convex body in X is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if X admits an equivalent locally uniformly convex norm, with related equivalences for Fréchet smooth approximations when X is reflexive or separable and Asplund.

What carries the argument

The Hausdorff metric on the collection of convex bodies, used together with renormings of X that produce local uniform convexity, Fréchet differentiability, uniform convexity, or uniform smoothness.

If this is right

  • If X admits an equivalent locally uniformly convex norm then every convex body can be approximated by locally uniformly convex bodies.
  • Every convex body can be approximated by Fréchet smooth convex bodies whenever the dual norm is locally uniformly convex.
  • If X is reflexive then every convex body can be approximated by both locally uniformly convex and Fréchet smooth convex bodies.
  • For separable X, approximation by both locally uniformly convex and Fréchet smooth bodies holds if and only if X is Asplund.
  • Uniform approximation by uniformly convex bodies is possible precisely when X is superreflexive, and the same holds for uniform smoothness.

Where Pith is reading between the lines

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

  • In spaces such as c0 or L1 that fail to be superreflexive, uniform approximation by uniformly convex bodies must be impossible.
  • The separable Asplund case suggests that good approximations by smooth and convex bodies can occur without full reflexivity.
  • The equivalences supply a new way to verify superreflexivity by checking approximation properties in explicit examples.
  • Similar equivalences might hold for other quantitative moduli of convexity and smoothness beyond the uniform case.

Load-bearing premise

The paper rests on the known affirmative answer to the original Klee problem in the strict convexity and Gâteaux smoothness setting.

What would settle it

A concrete example of a non-superreflexive Banach space in which every convex body can nevertheless be uniformly approximated by uniformly convex bodies would falsify the main equivalence.

read the original abstract

In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and G\^{a}teaux smoothness. In this paper, we continue to study this problem in strong senses, such as local uniform convexity, uniform convexity, Fr\'{e}chet smoothness and uniform smoothness. As a result, we show (1) Every convex body in a Banach space $X$ is approximated by locally uniformly convex bodies with respect to the Hausdorff metric if and only if $X$ admits an equivalent locally uniformly convex norm; (2) Every convex body in $X$ can be approximated by Fr\'echet smooth convex bodies if $X$ admits an equivalent norm so that its dual norm is locally uniformly convex on $X^*$; 3. Every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if $X$ is reflexive; 4. If $X$ is separable, then every convex body in $X$ can be approximated by both locally uniformly convex and Fr\'{e}chet smooth convex bodies if and only if $X$ is an Asplund space; (5) the following statements are equivalent: A. $X$ is super reflexive; B. Every convex body in $X$ can be uniformly approximated by uniformly convex bodies; C. Every convex body in $X$ can be uniformly approximated by uniformly smooth convex bodies; D. Every convex body in $X$ can be uniformly approximated by both uniformly convex and uniformly smooth convex bodies.

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

0 major / 3 minor

Summary. The manuscript strengthens Klee's convex body problem in Banach spaces by establishing equivalences and implications between approximability of convex bodies (w.r.t. the Hausdorff metric) by bodies with enhanced convexity/smoothness properties and renorming properties of the space. Building on the Cheng-Jiang-Yuan affirmative solution for strict convexity and Gâteaux smoothness, it proves: (1) every convex body approximable by locally uniformly convex bodies iff X admits an equivalent locally uniformly convex norm; (2) approximable by Fréchet smooth bodies if X admits an equivalent norm whose dual is locally uniformly convex; (3) approximable by both if X is reflexive; (4) for separable X, approximable by both iff X is Asplund; (5) X superreflexive iff every convex body is uniformly approximable by uniformly convex bodies, by uniformly smooth bodies, or by both.

Significance. If the derivations hold, the paper supplies a coherent set of characterizations that link geometric approximation questions directly to classical renorming theorems (local uniform convexity, Asplund property, superreflexivity). The equivalences in (5) are particularly clean and falsifiable; the one-way implications in (2) and (3) are consistent with known duality and reflexivity facts. The work properly credits the foundational Cheng-Jiang-Yuan result and employs only standard tools (Hausdorff metric, Minkowski sums with small balls).

minor comments (3)
  1. [Abstract] Abstract item (2): the statement is one-directional ('if'); the manuscript should explicitly state whether the converse is known to fail or is left open, and cite the relevant renorming obstruction if it exists.
  2. [Preliminaries] The preliminary section on the Hausdorff metric and gauge functionals should include a short self-contained paragraph recalling the precise definition and the fact that uniform approximation is equivalent to the existence of equivalent norms whose unit balls approximate the given body; this would make the 'only if' directions in (1) and (4) easier to follow without external references.
  3. [Theorem corresponding to item (4)] Item (4): the separability hypothesis is standard for Asplund characterizations, but the manuscript should briefly note why the 'only if' direction requires separability while the 'if' direction does not.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our results, as well as the recommendation for minor revision. The report correctly identifies the main equivalences and implications linking convex body approximation to renorming properties.

Circularity Check

1 steps flagged

Minor self-citation to base Klee result; derivations use independent standard renorming facts

specific steps
  1. self citation load bearing [Abstract, paragraph 1]
    "In a recent article, Cheng, Jiang and Yuan gave an affirmative answer to Klee's convex bodies problem of Banach spaces in the sense of strict convexity and Gâteaux smoothness. In this paper, we continue to study this problem in strong senses..."

    The equivalences (1)-(5) are stated to rest on the cited affirmative answer; however the citation is to prior work and the current derivations invoke only standard facts about the Hausdorff metric and equivalent norms, so the self-citation is minor and not internally circular.

full rationale

The paper cites Cheng-Jiang-Yuan (overlapping authors) for the strict-convexity/Gâteaux base case and then derives the listed equivalences via Hausdorff-metric approximations, Minkowski sums with small balls, and standard renorming theorems. These steps are logical implications from norm properties and do not reduce by construction to the cited result or to any fitted parameter. The self-citation is therefore not load-bearing for the new strengthened claims, which remain externally falsifiable via the usual characterizations of super-reflexivity, Asplund spaces, and uniform convexity/smoothness.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard axioms of Banach space theory (completeness, norm axioms, duality) and the Hausdorff metric on the space of convex bodies; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Every Banach space admits equivalent norms; the Hausdorff metric metrizes the space of closed convex bodies.
    Invoked implicitly when stating approximation results and equivalences with renorming.
  • standard math Reflexivity, Asplund property, and superreflexivity are well-defined and have the usual characterizations via norms and duals.
    Used in items (3), (4), (5) of the abstract.

pith-pipeline@v0.9.1-grok · 5841 in / 1489 out tokens · 17363 ms · 2026-06-27T20:33:31.845237+00:00 · methodology

discussion (0)

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Reference graph

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