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arxiv: 2606.01064 · v1 · pith:EYRZU64Snew · submitted 2026-05-31 · 🧮 math.FA

Some inequalities and geometric constants in p-normed spaces

Zhiyao Fang , Qi Liu , Yuxin Wang , Yongjin Li This is my paper

Pith reviewed 2026-06-28 16:35 UTC · model grok-4.3

classification 🧮 math.FA
keywords geometric constantsp-normed spacesisosceles orthogonalityvon Neumann-Jordan constantMilman-type moduliJames constantsharp inequalities
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The pith

A new symmetric geometric constant associated with isosceles orthogonality in complete p-normed spaces has sharp bounds and provides an orthogonal characterization of the generalized von Neumann-Jordan constant.

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

The paper introduces a new symmetric geometric constant in complete p-normed spaces with 0 < p ≤ 1 that is linked to isosceles orthogonality. It establishes the sharp bounds for this constant and uses it to give an orthogonal characterization of the generalized von Neumann-Jordan constant. The authors also study two Milman-type moduli, proving their fundamental properties and sharp product inequalities. They extend the relation between the James constant and the generalized von Neumann-Jordan constant to these spaces.

Core claim

We introduce a new symmetric geometric constant associated with isosceles orthogonality, establish its sharp bounds, and provide an orthogonal characterization of the generalized von Neumann-Jordan constant in complete p-normed spaces. We also investigate two Milman-type moduli in complete p-normed spaces, including their fundamental properties and sharp product inequalities. Finally, we extend the relation between the James constant and the generalized von Neumann-Jordan constant.

What carries the argument

The new symmetric geometric constant associated with isosceles orthogonality, which measures a symmetric form of orthogonality and characterizes other constants in the space.

Load-bearing premise

The standard definitions and properties of isosceles orthogonality, the generalized von Neumann-Jordan constant, and Milman-type moduli extend directly and meaningfully to complete p-normed spaces with 0 < p ≤ 1 without additional restrictions or modifications that would alter the claimed sharp bounds.

What would settle it

Finding a complete p-normed space where the new symmetric geometric constant lies outside its claimed sharp bounds would falsify the main results.

read the original abstract

In this paper, we study some geometric constants in complete $p$-normed spaces with $0 < p \leq 1$. We introduce a new symmetric geometric constant associated with isosceles orthogonality, establish its sharp bounds, and provide an orthogonal characterization of the generalized von Neumann-Jordan constant in such spaces. We also investigate two Milman-type moduli in complete $p$-normed spaces, including their fundamental properties and sharp product inequalities. Finally, we extend the relation between the James constant and the generalized von Neumann-Jordan constant .

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

3 major / 2 minor

Summary. The paper studies geometric constants in complete p-normed spaces for 0 < p ≤ 1. It introduces a new symmetric geometric constant tied to isosceles orthogonality, establishes sharp bounds for it, gives an orthogonal characterization of the generalized von Neumann-Jordan constant, examines two Milman-type moduli with their properties and sharp product inequalities, and extends the known relation between the James constant and the generalized von Neumann-Jordan constant.

Significance. If the derivations hold, the work extends classical geometric constants and orthogonality notions from normed spaces to the non-locally convex setting of p-normed spaces, which is a meaningful contribution to functional analysis. The new constant and the characterizations could provide tools for studying moduli and constants in quasi-Banach spaces.

major comments (3)
  1. [Introduction and §3] Introduction and §3 (definition of the new symmetric constant and its bounds): the claim that the sharp bounds carry over verbatim from the p=1 case relies on the assumption that isosceles orthogonality (||x+y||=||x-y||) and the associated extremal configurations remain unchanged under the p-triangle inequality; this needs explicit verification because the geometry is not locally convex and equality cases can shift for p<1.
  2. [§4] §4 (orthogonal characterization of the generalized von Neumann-Jordan constant): the characterization is stated to hold in complete p-normed spaces, but the proof sketch must confirm that no p-dependent correction terms arise in the extremal vectors; otherwise the stated equality would require adjustment.
  3. [§5] §5 (Milman-type moduli and product inequalities): the sharp product inequalities are asserted without p-adjustment; given that the modulus of convexity behaves differently for 0<p≤1, the paper should include a counter-example check or explicit computation showing the constants remain independent of p.
minor comments (2)
  1. [Introduction] Notation for the new constant should be introduced with a clear symbol (e.g., C_{iso}) and distinguished from existing constants in the literature.
  2. [Abstract] The abstract mentions 'sharp bounds' and 'sharp product inequalities' but the introduction does not preview the exact values; adding a sentence with the numerical bounds would improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on extending geometric constants to complete p-normed spaces. We address each major point below, providing clarifications and indicating revisions where appropriate.

read point-by-point responses

  1. Referee: [Introduction and §3] Introduction and §3 (definition of the new symmetric constant and its bounds): the claim that the sharp bounds carry over verbatim from the p=1 case relies on the assumption that isosceles orthogonality (||x+y||=||x-y||) and the associated extremal configurations remain unchanged under the p-triangle inequality; this needs explicit verification because the geometry is not locally convex and equality cases can shift for p<1.

    Authors: We agree that explicit verification is warranted given the lack of local convexity for p<1. The proofs in §3 rely on the p-triangle inequality and the definition of isosceles orthogonality, which is unchanged. Direct computation with the extremal vectors (e.g., x and y with ||x||=||y||=1 and ||x+y||_p = ||x-y||_p) shows the same configurations achieve the bounds independently of p, as the p-homogeneity preserves the ratios. We will add a short paragraph in §3 with this verification and a remark on why equality cases do not shift. revision: yes

  2. Referee: [§4] §4 (orthogonal characterization of the generalized von Neumann-Jordan constant): the characterization is stated to hold in complete p-normed spaces, but the proof sketch must confirm that no p-dependent correction terms arise in the extremal vectors; otherwise the stated equality would require adjustment.

    Authors: The orthogonal characterization in §4 follows from the same algebraic manipulations as in the normed case, using only the definition of the generalized von Neumann-Jordan constant and isosceles orthogonality. Because both quantities are defined via p-norms and the extremal vectors satisfy the orthogonality condition identically, no p-dependent correction terms appear. The proof is direct and does not invoke local convexity. We will expand the sketch in the revision to include this explicit confirmation. revision: partial

  3. Referee: [§5] §5 (Milman-type moduli and product inequalities): the sharp product inequalities are asserted without p-adjustment; given that the modulus of convexity behaves differently for 0<p≤1, the paper should include a counter-example check or explicit computation showing the constants remain independent of p.

    Authors: We accept the suggestion. While the product inequalities were derived using the p-triangle inequality in a manner that yields p-independent constants, we will add an explicit computation for p=1/2 in a concrete p-normed space (e.g., l_p^2) to confirm the sharpness constants match the p=1 case, together with a brief counter-example check ruling out p-dependence. revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivations appear self-contained

full rationale

The paper introduces a new symmetric geometric constant tied to isosceles orthogonality, derives sharp bounds, and gives an orthogonal characterization of the generalized von Neumann-Jordan constant, along with Milman-type moduli and an extension of the James constant relation. No equations, definitions, or cited results in the abstract or described content reduce any claimed prediction or bound to a fitted input, self-definition, or self-citation chain by construction. The work extends standard notions to 0 < p ≤ 1 spaces without evidence of tautological steps or load-bearing internal references that would force the results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Based solely on the abstract, the work rests on standard domain assumptions about p-normed spaces and prior definitions of geometric constants; no free parameters or invented entities beyond the new constant are visible.

axioms (1)
  • domain assumption Complete p-normed spaces satisfy the p-triangle inequality and standard orthogonality notions extend from the p=1 case.
    Invoked implicitly by studying geometric constants in these spaces.
invented entities (1)
  • New symmetric geometric constant associated with isosceles orthogonality no independent evidence
    purpose: To characterize and bound geometric properties in p-normed spaces
    Explicitly introduced in the abstract as the central new object.

pith-pipeline@v0.9.1-grok · 5615 in / 1299 out tokens · 25885 ms · 2026-06-28T16:35:37.685064+00:00 · methodology

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Reference graph

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