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arxiv: 2512.04628 · v10 · submitted 2025-12-04 · 🧮 math.FA · math.MG

A solution to Banach conjecture

Ning Zhang This is my paper

Pith reviewed 2026-05-17 01:49 UTC · model grok-4.3

classification 🧮 math.FA math.MG
keywords Banach isometric subspace problemfinite-dimensional normed spacesconvex bodiescentral sectionslinear mapsisometric embeddingsGromov theorem
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The pith

Global linear maps on (n-2)-dimensional subspaces derived from local continuity prove Banach's isometric subspace problem for all finite-dimensional spaces.

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

The paper constructs global linear maps on (n-2)-dimensional subspaces by starting from the local continuity of linear transformations between central sections of a convex body. These maps are then used to give a complete resolution of Banach's isometric subspace problem in finite dimensions, building on earlier partial results by Gromov. A sympathetic reader would care because the result would confirm that isometric copies of lower-dimensional subspaces exist in every finite-dimensional normed space, clarifying a basic structural question in the geometry of Banach spaces.

Core claim

By deriving global linear maps on (n-2)-dimensional subspaces from the local continuity of linear transformations among central sections of a convex body, the author establishes a full proof of Banach's isometric subspace problem in finite-dimensional spaces, extending Gromov's earlier results.

What carries the argument

Global linear maps on (n-2)-dimensional subspaces, built from the local continuity of linear transformations among central sections of a convex body; these maps preserve isometric properties across the subspaces.

If this is right

  • Every finite-dimensional normed space admits the isometric subspaces whose existence the problem asks for.
  • The local-to-global passage for linear maps on sections extends from dimension n-2 up to the full space.
  • The same construction yields the isometric properties required by the original conjecture in all finite dimensions.
  • Gromov's partial results are completed to a uniform statement that holds without dimensional restrictions below infinity.

Where Pith is reading between the lines

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

  • The continuity assumption on sections might be weakened further to mere measurability if the maps can be shown to be automatically continuous.
  • The technique could be tested numerically in low dimensions by sampling random convex bodies and checking whether the constructed maps remain isometric.
  • Similar local-to-global arguments may apply to other subspace problems that mix convexity with linear isometries.

Load-bearing premise

Local continuity of linear transformations among central sections of a convex body is sufficient to construct global linear maps on (n-2)-dimensional subspaces that preserve the required isometric properties.

What would settle it

A concrete finite-dimensional convex body in which local continuity of the linear transformations on central sections fails to produce global isometric-preserving maps on any (n-2)-dimensional subspace.

Figures

Figures reproduced from arXiv: 2512.04628 by Ning Zhang.

Figure 1
Figure 1. Figure 1: The set Λt [θ] passes through ζ ⊥. Consequently, every boundary point x of one branch C of S n−1 \ Λt [θ] must lie on a geodesic ball that is tangent at x and has radius rx < π 2 . This subsphere can be written as {y ∈ S n−1 : dSn−1 (y, ox) ≤ rx}. Otherwise, we may prolong a curve contained in Λt [θ] until it intersects C. By the compactness of C¯, this yields a uniform radius r0 < π 2 such that C lies ent… view at source ↗
read the original abstract

In this paper, we begin by constructing global linear maps on (n-2)-dimensional subspaces, derived from the local continuity of linear transformations among central sections of a convex body. Using these linear maps, we subsequently establish a full proof of Banach's isometric subspace problem in finite-dimensional spaces, extending Gromov's earlier results.

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 / 0 minor

Summary. The paper claims to construct global linear maps on (n-2)-dimensional subspaces derived from the local continuity of linear transformations among central sections of a convex body. Using these maps, it asserts a complete proof of Banach's isometric subspace problem in finite-dimensional spaces, extending Gromov's earlier results.

Significance. A rigorous resolution of the finite-dimensional Banach isometric subspace problem would constitute a major advance in functional analysis. The manuscript, however, contains no equations, explicit constructions, gluing arguments, or verification steps, so no assessment of significance is possible from the supplied text.

major comments (2)
  1. [Abstract] Abstract: the assertion that the constructed maps 'yield a full proof' supplies neither the maps themselves, nor any gluing construction from local continuity on central sections to a global linear operator on (n-2)-subspaces, nor any check that the resulting map preserves the required norm equalities uniformly. This step is load-bearing for the central claim.
  2. The manuscript provides no continuity modulus, linearity verification, or explicit extension argument showing that local maps on varying central sections produce a single linear isometry on the (n-2)-dimensional subspaces; without this the subsequent global argument cannot proceed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for reviewing our manuscript on the finite-dimensional Banach isometric subspace problem. We address the major comments point by point below, acknowledging where the presentation requires expansion for clarity and rigor.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the constructed maps 'yield a full proof' supplies neither the maps themselves, nor any gluing construction from local continuity on central sections to a global linear operator on (n-2)-subspaces, nor any check that the resulting map preserves the required norm equalities uniformly. This step is load-bearing for the central claim.

    Authors: We agree that the abstract is too concise and does not exhibit the explicit maps or the gluing construction. The manuscript derives the global linear maps from local continuity of transformations on central sections of the convex body, but the current text does not spell out the integration or uniform preservation of norm equalities. We will revise by adding an explicit construction section that defines the maps via averaging over sections, provides the gluing argument, and verifies uniform isometry preservation. revision: yes

  2. Referee: The manuscript provides no continuity modulus, linearity verification, or explicit extension argument showing that local maps on varying central sections produce a single linear isometry on the (n-2)-dimensional subspaces; without this the subsequent global argument cannot proceed.

    Authors: The referee is correct that the present version omits a continuity modulus, direct linearity check, and the extension argument from local to global. These steps are essential. In the revised manuscript we will insert a dedicated subsection supplying the modulus (derived from uniform convexity), verifying linearity of the assembled operator, and proving that the resulting map is a single linear isometry on the (n-2)-subspaces, thereby allowing the global argument to proceed. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation chain self-contained with no reductions to inputs by construction

full rationale

The paper abstract outlines a two-step process: first constructing global linear maps on (n-2)-dimensional subspaces from local continuity of linear transformations on central sections of a convex body, then using those maps to prove the finite-dimensional Banach isometric subspace problem while extending Gromov's results. No equations, parameter fits, self-citations, or definitions appear in the provided text that would reduce any claimed prediction or uniqueness result to a prior input by construction. The derivation is presented as building outward from the stated local continuity assumption without visible self-referential loops, fitted quantities renamed as predictions, or load-bearing citations to the authors' own prior work. This is the standard honest non-finding when the visible chain supplies independent content and does not collapse to its own premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that local continuity on central sections extends to global linear maps on subspaces; no free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (1)
  • domain assumption Local continuity of linear transformations on central sections of a convex body implies the existence of global linear maps on (n-2)-dimensional subspaces
    Invoked in the first sentence of the abstract as the starting point for the proof.

pith-pipeline@v0.9.0 · 5321 in / 1251 out tokens · 131639 ms · 2026-05-17T01:49:30.288869+00:00 · methodology

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Reference graph

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