Chebyshev centers and radius of the set of permutons

Bal\'azs Maga

arxiv: 2602.02889 · v2 · pith:PJ5MWEUNnew · submitted 2026-02-02 · 🧮 math.CO · math.MG· math.PR

Chebyshev centers and radius of the set of permutons

Bal\'azs Maga This is my paper

Pith reviewed 2026-05-21 13:19 UTC · model grok-4.3

classification 🧮 math.CO math.MGmath.PR

keywords permutonsChebyshev radiusChebyshev centersrectangular distance1/2-periodic measuresmetric geometryuniform marginals

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The pith

The set of permutons under rectangular distance has Chebyshev radius exactly 1/4, with centers precisely the 1/2-periodic measures in each coordinate.

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

This paper examines the geometry of the set of permutons using the rectangular distance. It establishes that the smallest radius sufficient to cover every permuton from some central point is 1/4. The only points that achieve this covering radius are the permutons that repeat their structure with period 1/2 along both axes. The work also identifies which permutons sit exactly at distance 1/4 from any given center. These facts describe the shape and extent of the entire collection of such measures in this metric.

Core claim

We study the metric geometry of the set of permutons under the rectangular distance d_square. We determine the Chebyshev radius to be 1/4 and characterize all Chebyshev centers: a permuton is a center if and only if it is 1/2-periodic in each coordinate. We also describe permutons that attain the extremal distance 1/4 from a given center.

What carries the argument

The rectangular distance d_square, which turns the set of permutons into a metric space whose Chebyshev radius and centers are then computed directly.

If this is right

Every permuton lies at distance at most 1/4 from any 1/2-periodic permuton.
A permuton fails to be a Chebyshev center precisely when it lacks 1/2-periodicity in at least one coordinate.
For each qualifying center there exist permutons realizing the distance 1/4.
The covering radius of the full set is attained and equals 1/4.

Where Pith is reading between the lines

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

Half-periodic permutons could serve as canonical reference points when approximating large random permutations by simpler objects.
The same radius-and-center question for other natural distances on permutons might produce different periodicities or different numerical values.
Empirical sampling of random permutations could test whether their induced measures cluster near the half-periodic class.

Load-bearing premise

That the rectangular distance turns the set of permutons into a metric space in which the Chebyshev radius exists and equals 1/4 exactly when the center is 1/2-periodic, without further restrictions from the uniform-marginal condition.

What would settle it

Exhibit one concrete permuton that is not 1/2-periodic yet covers the whole set with radius strictly less than 1/4, or exhibit one 1/2-periodic permuton that leaves at least one other permuton more than 1/4 away.

read the original abstract

We study the metric geometry of the set of permutons under the rectangular distance $d_{\square}$. We determine the Chebyshev radius to be 1/4 and characterize all Chebyshev centers: a permuton is a center if and only if it is 1/2- periodic in each coordinate. We also describe permutons that attain the extremal distance 1/4 from a given center.

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.

Desk Editor's Note private letter to a colleague

The paper computes the Chebyshev radius of permutons under rectangular distance as exactly 1/4 and characterizes the centers as the 1/2-periodic ones.

read the letter

The main result is that the Chebyshev radius is 1/4 and a permuton is a center if and only if it is 1/2-periodic in each coordinate. They also describe the extremal permutons at distance 1/4 from a center. This gives a concrete number and a clean if-and-only-if statement for this metric on the space of permutons, which is not something I had seen stated before in the literature on permuton geometry. The rectangular distance is a reasonable choice here because it directly measures discrepancies over rectangles while respecting the uniform marginals. The paper does a straightforward job stating the claim and presumably deriving the periodicity condition from the sup definition of the distance. The result supplies a specific geometric fact that could be referenced when working with distances or coverings in this space. The soft spot is the interaction between 1/2-periodicity and the uniform marginal constraints. The stress-test note flags that the marginal conditions fix measures on strips, so it is not automatic that periodicity is compatible without forcing deviations or that non-periodic permutons could still achieve the same covering radius. The abstract presents the characterization as holding, but the proof needs to show explicitly that the marginals do not create extra centers or block the periodic ones. If that step is handled cleanly, the claim stands; otherwise the iff part weakens. The math looks like standard metric geometry applied to measures with fixed marginals, with no obvious circularity. This is for readers working on metric properties of permutons or combinatorial geometry who want an explicit constant rather than existence results. It deserves a serious referee because the claim is sharp and the characterization is usable if the details check out. Recommendation: send it for review.

Referee Report

2 major / 2 minor

Summary. The paper studies the metric geometry of the set of all permutons (probability measures on [0,1]^2 with uniform marginals) under the rectangular distance d_□. It claims to determine that the Chebyshev radius of this set is exactly 1/4 and to characterize the Chebyshev centers as precisely the 1/2-periodic permutons in each coordinate. The manuscript also describes the permutons that attain distance 1/4 from a given center.

Significance. If the central claims hold, the result gives an explicit and complete description of the smallest enclosing ball in the space of permutons under d_□. This supplies concrete geometric information about an important compact metric space arising in combinatorial probability and optimal transport, and the explicit radius together with the periodicity characterization may serve as a reference point for related extremal problems.

major comments (2)

[§3 (proof of the center characterization)] The proof of the 'if and only if' characterization (presumably Theorem 1.2 or the main result in §3) must verify that every 1/2-periodic measure with uniform marginals is indeed a center and, conversely, that no non-periodic permuton can serve as a center. The uniform-marginal constraints impose linear conditions on the measures of rectangles; it is not immediate that these conditions are automatically satisfied by arbitrary 1/2-periodic measures or that they exclude all non-periodic candidates from achieving radius 1/4. A concrete check that the periodicity condition is compatible with the marginals and that the distance definition does not admit additional centers is required.
[§2 (radius computation) and §4 (extremal-distance description)] The argument that the radius is exactly 1/4 relies on exhibiting a center and showing that no smaller radius works. The lower bound must be established by exhibiting two permutons whose d_□ distance is 1/4; the upper bound requires showing that every permuton lies within distance 1/4 of any 1/2-periodic center. Both directions should be checked against the marginal constraints to ensure no hidden restrictions reduce the attainable radius or enlarge the set of centers.

minor comments (2)

[Introduction] Notation for the rectangular distance d_□ should be introduced with an explicit formula (supremum of |μ(R) - λ(R)| over axis-aligned rectangles R) before it is used in statements.
[§1] The definition of 1/2-periodicity for a measure on [0,1]^2 should be stated precisely (e.g., invariance under the map (x,y) ↦ (x+1/2 mod 1, y+1/2 mod 1) or the corresponding condition on rectangle measures).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying points where the proofs of the main results would benefit from additional explicit verifications. We address each major comment below and will incorporate the suggested clarifications into a revised version of the paper.

read point-by-point responses

Referee: [§3 (proof of the center characterization)] The proof of the 'if and only if' characterization (presumably Theorem 1.2 or the main result in §3) must verify that every 1/2-periodic measure with uniform marginals is indeed a center and, conversely, that no non-periodic permuton can serve as a center. The uniform-marginal constraints impose linear conditions on the measures of rectangles; it is not immediate that these conditions are automatically satisfied by arbitrary 1/2-periodic measures or that the distance definition does not admit additional centers. A concrete check that the periodicity condition is compatible with the marginals and that the distance definition does not admit additional centers is required.

Authors: We agree that the compatibility of 1/2-periodicity with the uniform-marginal conditions and the exclusion of non-periodic centers require explicit verification. In the revised manuscript we will add a short preliminary result in §3 showing that any 1/2-periodic probability measure on [0,1]^2 whose marginals are Lebesgue measure is automatically a permuton, and we will then compute the rectangular distance d_□ directly from an arbitrary permuton to such a center, confirming that the supremum is exactly 1/4. For the converse direction we will strengthen the argument by constructing, for any non-1/2-periodic permuton, an explicit test permuton (a suitable two-step measure respecting the marginal constraints) whose rectangular distance exceeds 1/4. These additions will make the linear constraints on rectangle measures fully transparent. revision: yes
Referee: [§2 (radius computation) and §4 (extremal-distance description)] The argument that the radius is exactly 1/4 relies on exhibiting a center and showing that no smaller radius works. The lower bound must be established by exhibiting two permutons whose d_□ distance is 1/4; the upper bound requires showing that every permuton lies within distance 1/4 of any 1/2-periodic center. Both directions should be checked against the marginal constraints to ensure no hidden restrictions reduce the attainable radius or enlarge the set of centers.

Authors: We will expand §2 to include an explicit pair of permutons attaining d_□ = 1/4: the uniform measure supported on the main diagonal and the measure supported on the union of the two off-diagonal squares [0,1/2]×[1/2,1] and [1/2,1]×[0,1/2], both of which have uniform marginals. In §4 we will verify the upper bound by integrating the discrepancy over all axis-aligned rectangles while enforcing the marginal uniformity at each step; the resulting estimate shows that the distance is at most 1/4 for every permuton, with no reduction caused by the marginal constraints. These concrete checks will confirm that the radius is precisely 1/4 and that the set of centers is not enlarged by the marginal conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: radius and center characterization derived from metric geometry

full rationale

The paper computes the Chebyshev radius of the permuton space under d_□ as 1/4 and proves an iff characterization of centers via 1/2-periodicity. No quoted step reduces a claimed prediction or uniqueness result to a fitted parameter, self-definition, or load-bearing self-citation. The uniform-marginal constraint is treated as part of the ambient space rather than an after-the-fact restriction that would force the result by construction. The derivation is therefore self-contained against the stated metric and measure-theoretic definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard definition of permutons as doubly stochastic measures and on the rectangular distance being a well-defined metric; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The rectangular distance d_square is a metric on the set of permutons.
This metric is the one used to define Chebyshev radius and centers in the abstract.

pith-pipeline@v0.9.0 · 5584 in / 1230 out tokens · 57584 ms · 2026-05-21T13:19:52.128687+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. The Chebyshev radius of Perm in d□ is 1/4, and a permuton is a Chebyshev center if and only if it is 1/2-periodic.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 3.1 ... equivalent characterizations of 1/2-periodicity ... for any toric rectangle R with h+w=1, μ(R)≤1/4.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.