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bibkey: mannamandalpaulsain2025improvement
authors: J. Manna; K. Mandal; K. Paul; D. Sain
year: 2025
title: "An improvement of the Blanco-Koldobsky-Turnšek characterization of isometries"
doi: null
url: https://arxiv.org/abs/2511.20345v1
claim: "The final Open Question asks whether preservation at any chosen n independent extreme points forces a bounded endomorphism of every n-dimensional polyhedral Banach space to be a scalar multiple of an isometry."
strata_touched:
  - D5/S3/Analytic/BanachGeometry/PolyhedralBJRefutation
license: citation-only
triage: anchor

Verified locator

Canonical versioned source: https://arxiv.org/abs/2511.20345v1.

Jayanta Manna, Kalidas Mandal, Kallol Paul and Debmalya Sain, An improvement of the Blanco-Koldobsky-Turnšek characterization of isometries, arXiv:2511.20345v1, submitted 2025-11-25. The versioned author text ends with the following question, with rendered mathematical symbols transcribed and whitespace normalized:

Open Question: In an n-dimensional polyhedral Banach space, does the preservation of Birkhoff-James orthogonality by a bounded linear operator at any n linearly independent extreme points necessarily imply that the operator is a scalar multiple of an isometry?

The standing field is real or complex (S1.p6.1). Birkhoff–James orthogonality is the inequality $\Vert x\Vert \le \Vert x+\lambda y\Vert$ for every base-field scalar $\lambda$ (S1.p7.1). Local preservation quantifies every direction $y$ (S1.p8.1). The proper-subset discussion immediately before the question (S2.p44.1) distinguishes a chosen independent tuple from a requirement to preserve at every extreme point. The final question has no norm-one, nonzero-operator, regularity or smoothness restriction. The preceding all-extreme-point theorem does not supply such a restriction to the question.

Polyhedrality is implemented by finiteness of the actual real-convex extreme points of the closed unit ball. This agrees with the explicit finite-dimensional definition in the authors’ arXiv:2407.08900v1, S1.p9.1, and arXiv:2504.11161v2, S1.p2. The source conclusion is a single scalar times a linear isometry, including scalar zero.

Bounded comparisons

The complete numbered statements and explicit examples of arXiv:2504.11161v2, revised 2026-02-07, and arXiv:2501.02714v1, submitted 2025-01-06, have been compared with this question. The entire proof and both witness constructions of the latter’s Theorem 2.33 are included in that comparison. These source readings establish the following bounded distinctions:

• Theorems 2.19 and 2.27 and Corollary 2.20 of 2504.11161v2 require every extreme point or every point of a specified smoothness order. They do not replace this test set by an arbitrary independent tuple.
• Lemma 2.16 and Theorem 2.18 of that paper extend preservation along a tested face. In the example below $[p,q]$ is one face and its image $[p,-r]$ is another. This extension does not reach the omitted vertex $r$.
• Remark 2.15 gives a Hilbert-space projection example, whose ball is not polyhedral. The explicit ${\ell }_p^n$, $1<p<\infty$, $p\ne 2$, examples in Theorem 2.33 of 2501.02714v1 likewise have infinitely many extreme points. They are the arbitrary-Banach-space negative phenomenon cited after the target question, and do not satisfy its polyhedral premise.
• Corollary 2.29 of 2501.02714v1 gives bijectivity for two independent extreme tests in a polyhedral plane. Bijectivity does not imply the scalar-isometry conclusion. The operator below is invertible.
• The existing sup-norm result characterizes extreme-point K-sets only on ${\ell }_{\infty }^n$. Norm equivalence does not preserve BJ orthogonality; it supplies no bridge to an irregular hexagonal norm.

No inspected theorem or explicit example in these two versioned full texts supplies this whole polyhedral resolution. This is a bounded comparison, not a global absence, priority or present-openness certificate. No new source requests are part of the native implementation.

Native counterexample and correspondence

On ${\mathbb{R}}^2$ put

$$N(a,b)=\max \{|a|,|a+b|,|a/2+b|\},T(a,b)=(a+b,-3b/2).$$

The native carrier is the submodule $w_2=w_1-w_0/2$ in the three-dimensional sup-norm space, with $(a,b)\mapsto (a,a+b,a/2+b)$. Its inherited norm is exactly $N$, its real dimension is two, and it is complete. The actual unit ball has finitely many extreme points, contained in $\{\pm p,\pm q,\pm r\}$, where $p=(1,0)$, $q=(0,1)$ and $r=(-1,3/2)$. The native proof establishes that $p,q$ are extreme unit vectors and an independent pair, constructs $T$ as a continuous linear endomorphism, and proves full local BJ preservation at both selected vectors for every real direction and scalar. Norms $N(Tp)=1$ and $N(Tr)=2$, with $N(p)=N(r)=1$, contradict every equality $T=cU$ with native linear isometry $U$.

PolyhedralBJRefutation.claim is a closed proposition, the conjunction of the universal real and complex assertions. Its unique designated result is literally Not claim. The real instance refutes that conjunction; the claim has no free counterexample assumption and no missing realization premise. This settles one universal question, without a classification of polyhedral spaces or a determination of their K-numbers.

Scope

The prospective native-implementation registration is issue #19309. The ordinary counterexample and the bounded comparisons preceded that issue; they are not retroactively preregistered. Journal-version equivalence and later unindexed or unpublished work are unverified. The cited source identities and anchors are the limits of the literature statements here.