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arxiv: 2509.14843 · v2 · submitted 2025-09-18 · 🧮 math.OC

On the Multiary Algebraic Formulation of an Idempotent Symmetric Limit Convex Structure

Walter Briec This is my paper

Pith reviewed 2026-05-18 16:17 UTC · model grok-4.3

classification 🧮 math.OC
keywords B-convexityPainlevé-Kuratowski limitalgebraic formulationidempotent structuremultiary operationconvex polytopesseparation theorem
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The pith

The Painlevé-Kuratowski limits of linear convex polytopes admit a multiary algebraic formulation using an idempotent non-associative structure that extends prior binary versions of B-convexity to any finite number of points.

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

This paper supplies an algebraic description for the polytopes that appear as limits in the original definition of B-convexity. B-convexity had been introduced through upper limits of sequences of linear convex sets without an explicit algebraic expression for those limits. The new multiary operation is idempotent and non-associative, generalizing an earlier binary algebraic account of B-convexity to arbitrary collections of points. The resulting structure is shown to differ from the idempotent symmetrical convex structure introduced in related work. The algebraic view also produces a separation theorem for convex sets in Euclidean space by approximating them with the limiting polytopes.

Core claim

The limiting polytopes obtained from the Painlevé-Kuratowski limit of linear convexities possess an algebraic formulation that is idempotent and non-associative. This formulation yields a multiary version of B-convexity defined on an arbitrary number of points and demonstrates that the structure fails to satisfy the idempotent symmetrical convex structure of earlier references. A general separation result in R^n follows by approximating convex sets with these polytopes, and the external representation of the polytopes is clarified.

What carries the argument

The idempotent non-associative multiary algebraic operation that describes the limiting polytopes arising from sequences of linear convex sets.

If this is right

  • B-convexity can now be expressed algebraically for any finite collection of points rather than only pairs.
  • The limiting polytopes do not obey the idempotent symmetrical convex structure introduced in related references.
  • Convex sets in R^n can be approximated by these polytopes to obtain a general separation theorem.
  • The external representation of the polytopes and their structural properties receive direct clarification.

Where Pith is reading between the lines

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

  • The multiary operation may support new algorithms for optimization problems that require simultaneous handling of several points under B-convexity.
  • Links between this non-associative structure and other algebraic models of convexity in the literature could be examined for unification.
  • The observed failure of symmetry invites re-examination of how limit processes affect algebraic properties of convexity.

Load-bearing premise

The Painlevé-Kuratowski limits of linear convex polytopes admit an algebraic description that remains idempotent and non-associative while differing from the symmetrical convex structure defined in prior work.

What would settle it

An explicit computation for three points showing that the proposed multiary operation produces a different set than the Painlevé-Kuratowski upper limit of the corresponding linear convex polytopes would refute the claimed algebraic formulation.

Figures

Figures reproduced from arXiv: 2509.14843 by Walter Briec.

Figure 3.2
Figure 3.2. Figure 3.2: Limit of Cop (x, y). Moreover, for all x, y ∈ R n such that I(x, y) ̸= ∅, there is a sequence of indexes {im}m∈[n(x,y)] ⊂ I(x, y) such that setting t ⋆ i0 = 0, t ⋆ in(x,y)+1 = +∞ and t ⋆ im = − xim yim for all m ∈ [n(x, y)]: γ (p) [PITH_FULL_IMAGE:figures/full_fig_p008_3_2.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: 1: 2-dimensional φp-polytope and intermediate points [PITH_FULL_IMAGE:figures/full_fig_p010_3_2.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: The domain cl ⟨a, .⟩∞ ≤ c [PITH_FULL_IMAGE:figures/full_fig_p025_4_1.png] view at source ↗
read the original abstract

In [14], B-convexity was defined as an appropriate Painlev\'e-Kuratowski limit of linear convexities. More recently, an alternative algebraic formulation over the entire Euclidean vector space was proposed in [9] and [10]. The issue with the definition presented in [14] is that it was not developed from an algebraic perspective, but rather as the upper limit of a sequence of generalized convex polytopes whose form was not explicitly given. In this paper, we build on recent work and provide an algebraic formulation for these limiting polytopes. Consequently, we deduce a multiary algebraic form of B-convexity that involves an idempotent, non-associative algebraic structure, extending the formalism proposed in [9] to an arbitrary number of points. Among other things, we demonstrate that these limiting polytopes do not satisfy the idempotent symmetrical convex structure defined in [9]. In the context of this formalism, we derive a general separation result in Rn by approximating convex sets by polytopes. We conclude by clarifying some points regarding the external representation of polytopes proposed in [11] and analyze the structure of the polytopes that arise in this context.

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 provides an algebraic formulation for the limiting polytopes obtained as the Painlevé-Kuratowski limit of sequences of linear convex polytopes, thereby deducing a multiary algebraic expression for B-convexity. This expression is claimed to define an idempotent, non-associative operation that extends the binary algebraic framework of the author's prior work [9] to an arbitrary number of points. The manuscript further asserts that these limiting structures fail to satisfy the idempotent symmetrical convex structure introduced in [9], derives a separation theorem in R^n via polytope approximation of convex sets, and clarifies points concerning the external representation of polytopes from [11].

Significance. If the central equivalence between the proposed algebraic multiary operation and the set-theoretic PK-limit holds with both inclusions verified, the work would supply a concrete algebraic handle on B-convexity for higher arities, potentially enabling direct manipulation of combinations and supporting the derived separation result. The explicit contrast with the symmetric structure of [9] and the clarification on external representations are useful for situating the contribution within the author's sequence of papers on algebraic convexity.

major comments (3)
  1. [Section defining the multiary algebraic form (near the statement of the main algebraic expression for the limiting polyt] The central deduction that the constructed multiary algebraic expression recovers the Painlevé-Kuratowski limit of linear convex polytopes (both inclusions) is not accompanied by an explicit verification for arity greater than 2. The manuscript extends the binary case from [9] but does not supply the direct or inductive argument showing that the algebraic combinations coincide with the set-theoretic limit for arbitrary sequences whose PK-limit exists; this equivalence is load-bearing for the multiary extension and the subsequent algebraic properties.
  2. [Paragraph presenting the negative result on the symmetric structure] The claim that the limiting polytopes fail to satisfy the idempotent symmetrical convex structure of [9] requires a precise identification of the violated axiom together with a concrete counter-example or derivation. Without this, the negative result remains an assertion rather than a demonstrated distinction between the two formalisms.
  3. [Section on the separation result] The general separation result in R^n obtained by approximating convex sets by polytopes is stated but lacks a detailed proof or even a sketched argument showing how the algebraic formulation of the limit is used to pass from polytope separation to the general case. This step is central to the claimed utility in convex analysis.
minor comments (2)
  1. Notation for the multiary operation and the limiting polytopes should be introduced with explicit symbols and consistently used throughout; current presentation mixes descriptive phrases with implicit definitions.
  2. The abstract and introduction cite [9], [10], [11], and [14] heavily; a brief self-contained recap of the binary algebraic form from [9] would improve readability for readers unfamiliar with the prior sequence.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thorough review and valuable suggestions. We address each of the major comments in detail below and outline the revisions we plan to make to strengthen the manuscript.

read point-by-point responses

  1. Referee: The central deduction that the constructed multiary algebraic expression recovers the Painlevé-Kuratowski limit of linear convex polytopes (both inclusions) is not accompanied by an explicit verification for arity greater than 2. The manuscript extends the binary case from [9] but does not supply the direct or inductive argument showing that the algebraic combinations coincide with the set-theoretic limit for arbitrary sequences whose PK-limit exists; this equivalence is load-bearing for the multiary extension and the subsequent algebraic properties.

    Authors: We acknowledge that the verification for arities greater than 2 could be made more explicit. The construction in the manuscript is designed to generalize the binary case, but to address this, we will include in the revised manuscript an inductive proof on the arity. This proof will verify both inclusions: that the multiary algebraic operation applied to points in the limit set yields points in the PK-limit, and conversely, that points in the PK-limit can be obtained as algebraic combinations. This will be added near the statement of the main algebraic expression. revision: yes

  2. Referee: The claim that the limiting polytopes fail to satisfy the idempotent symmetrical convex structure of [9] requires a precise identification of the violated axiom together with a concrete counter-example or derivation. Without this, the negative result remains an assertion rather than a demonstrated distinction between the two formalisms.

    Authors: We agree that providing a concrete counterexample would better illustrate the distinction. In the revision, we will add a specific example, for instance in R^2 with a sequence of polytopes whose PK-limit violates the symmetry axiom of the idempotent symmetrical convex structure from [9]. We will identify the exact axiom violated and derive why the limiting structure does not satisfy it, thereby demonstrating the difference between the two formalisms. revision: yes

  3. Referee: The general separation result in R^n obtained by approximating convex sets by polytopes is stated but lacks a detailed proof or even a sketched argument showing how the algebraic formulation of the limit is used to pass from polytope separation to the general case. This step is central to the claimed utility in convex analysis.

    Authors: We thank the referee for pointing this out. The separation result is obtained by approximating general convex sets with polytopes and using the algebraic formulation for the limits of those polytopes. In the revised version, we will include a sketched argument: First, establish separation for polytopes using the multiary algebraic expression. Then, leverage the fact that any convex set can be approximated by polytopes in the Hausdorff metric, combined with the continuity of the Painlevé-Kuratowski limit, to extend the separation to general convex sets in R^n. This will explicitly show the utility of the algebraic approach. revision: yes

Circularity Check

1 steps flagged

Multiary B-convexity extension and negative result on symmetric structure both reduce to self-cited binary formalism in [9] without independent multiary verification against PK-limit

specific steps
  1. self citation load bearing [Abstract]
    "we build on recent work and provide an algebraic formulation for these limiting polytopes. Consequently, we deduce a multiary algebraic form of B-convexity that involves an idempotent, non-associative algebraic structure, extending the formalism proposed in [9] to an arbitrary number of points. Among other things, we demonstrate that these limiting polytopes do not satisfy the idempotent symmetrical convex structure defined in [9]."

    The multiary algebraic form is obtained by extending the author's prior binary formalism in [9]; the negative result on the symmetric structure is likewise a direct comparison to the definition given in that same self-citation. The step that equates the constructed algebraic expression to the actual Painlevé-Kuratowski limit for arbitrary arity therefore reduces to the self-cited framework rather than an independent check.

full rationale

The paper's core deduction starts from the PK-limit definition in [14] and supplies an algebraic expression for the limiting polytopes by extending the binary algebraic formulation already introduced in the author's own prior papers [9] and [10]. The multiary operation is then declared to be the B-convex combination, and non-associativity/idempotence are derived from that expression. The claim that the limiting polytopes fail the idempotent symmetrical convex structure is likewise a direct comparison to the definition in [9]. Because the equivalence of the new algebraic expression to the set-theoretic PK-limit (both inclusions) for arbitrary arity is not re-verified from first principles but obtained by construction from the self-cited binary case, the central results are load-bearing on the prior self-citation chain. The paper does contain an independent separation theorem and clarifications on external representations, preventing a higher score.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the prior definition of B-convexity as a Painlevé-Kuratowski limit and on the algebraic structures introduced in the author's earlier papers; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption B-convexity defined as Painlevé-Kuratowski upper limit of linear convexities
    Stated in the opening sentence referencing [14]
  • domain assumption Existence of an algebraic formulation of B-convexity over Euclidean space
    Referenced from [9] and [10]

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