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Near-maximal Minkowski asymmetry forces a convex body to lie within a dimension-free Banach-Mazur distance of the simplex, tight up to linear order in the deficit.

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2026-07-30 12:55 UTC pith:HKLMWJ3B

load-bearing objection Dimension-free stability near the simplex is real and usable; the matching lower bound for n≥3 leans on unpublished planar work, but the upper bounds and applications stand on their own. the 1 major comments →

arxiv 2607.27041 v1 pith:HKLMWJ3B submitted 2026-07-29 math.MG

Tight Stability Estimates Near the Simplex and Improved Bounds for the Diameter of the Banach-Mazur Compactum in Fixed Dimensions

Ren\'e Brandenberg , Bernardo Gonz\'alez Merino , Florian Grundbacher This is my paper
classification math.MG MSC 52A4052A21
keywords Banach-Mazur distanceMinkowski asymmetrystabilitysimplexLoewner ellipsoidBanach-Mazur compactumconvex bodies
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that any n-dimensional convex body whose Minkowski asymmetry is at least n minus a small error must sit at Banach-Mazur distance at most 1 plus that error plus a quadratic correction from the simplex. The bound no longer depends on dimension and is shown to be best possible up to the linear term by explicit constructions that lift planar trapezoids. The same estimate upgrades the known stability of the maximal distance to the Euclidean ball to optimal linear order and yields sharper numerical upper bounds on the diameter of the Banach-Mazur compactum in every fixed dimension. A supporting lemma answers a conjecture of Belloni and Freund by showing that every body contains a suitably translated and scaled copy of its minimal-volume circumscribed ellipsoid. Together these results turn a classical extremal fact about simplices into a quantitatively useful stability tool.

Core claim

If an n-dimensional convex body K satisfies s(K) ≥ n − ε for ε ∈ [0, 1), then its Banach-Mazur distance to the n-simplex is at most 1 + ε + ε²/(2(1 − ε)). The bound is independent of dimension, improves all earlier estimates, and is sharp up to the linear term: for every n and every ε there exist bodies realizing distance exactly 1 + ε.

What carries the argument

Asymmetry-point simplices (simplices whose vertices are simultaneous contact points of K and its maximally scaled reflection) together with an explicit polyhedral superset of K built from those vertices and n additional near-vertex points; covering the superset by a translate of the simplex via support-function averaging yields the distance bound.

Load-bearing premise

The exact distance formula for planar trapezoids and the claim that coning preserves Banach-Mazur distance to the simplex whenever that distance is at most 2 must both hold, since the higher-dimensional tightness examples are obtained by iterated coning of those trapezoids.

What would settle it

Exhibit a single convex body in some dimension n ≥ 3 whose Minkowski asymmetry equals n − ε yet whose Banach-Mazur distance to the simplex strictly exceeds 1 + ε + ε²/(2(1 − ε)), or show that every planar trapezoid of asymmetry s has distance to the triangle strictly larger than 3 − s.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Any geometric inequality whose equality case is uniquely the simplex immediately inherits a dimension-free stability estimate of linear order in the deficit of Minkowski asymmetry.
  • The Banach-Mazur diameter of the set of all n-dimensional convex bodies is at most n² − n + 1/2 + O(1/n), with explicit numerical values already improved for every n from 3 to 8.
  • Bodies at Banach-Mazur distance n − ε from the Euclidean ball lie at distance less than 1 + 2ε + 2ε²/(1 − 2ε) from the simplex whenever ε < 1/2.
  • Every convex body contains a translated copy of its Loewner ellipsoid scaled by 1/√(n s(K)), settling the Belloni-Freund conjecture in the non-concentric form.

Where Pith is reading between the lines

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

  • The same contact-point geometry that controls asymmetry may yield dimension-free stability for other classical functionals (Rogers-Shephard volume, affine isoperimetric deficit) once their equality cases are known to be simplices.
  • Because the bound is already tight up to the linear term, any further improvement must exploit both contact points and supporting hyperplanes simultaneously rather than either family alone.
  • The improved diameter bounds in low dimensions are now small enough that exhaustive numerical search for extremal pairs becomes feasible for n = 3 and n = 4.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper proves a dimension-free stability estimate for Minkowski asymmetry near its maximum: if s(K)≥n−ε for ε∈[0,1), then d_BM(K,Δ_n)≤1+ε+ε²/(2(1−ε)), improving Schneider’s earlier bound and shown optimal up to linear order in ε. As applications it derives a linear-order stability result for bodies nearly maximizing d_BM(·,B₂ⁿ), affirms a conjecture of Belloni–Freund by showing that every body contains a suitably translated copy of its Loewner ellipsoid scaled by 1/√(n s(K)), and obtains improved explicit upper bounds on the diameter of the Banach–Mazur compactum in fixed dimensions n≥3.

Significance. Removing the dimension dependence from the classical stability estimate for Minkowski asymmetry is a clear advance; the resulting bound is elementary, sharp up to the linear term, and immediately upgrades several earlier applications (Kobos, Hug–Schneider, diameter estimates). The self-contained proof of the upper bound via asymmetry-point simplices and an explicit polyhedral superset (Section 2) is technically solid. The affirmative answer to the Belloni–Freund conjecture (Lemma 4.1) and the concrete numerical improvements for the Banach–Mazur diameter in low dimensions are additional concrete contributions of lasting reference value.

major comments (1)
  1. [Section 3, Propositions 3.2–3.4 and the induction proof of tightness] The claim that Theorem 1.1 is “best possible up to the linear order” for all n≥2 rests on the existence half proved in Section 3. The base case (n=2) invokes Proposition 3.2 (d_BM(trapezoid,Δ₂)=3−s(K)), which is taken from the unpublished working manuscript [21]; the inductive step further relies on the concurrent preprint [22, Thm 1.4] for distance preservation under coning. Without a self-contained proof (or a published reference) of the planar identity, the existence statement for n≥3 is not fully established by the written arguments. Either include a short derivation of the trapezoid formula or qualify the tightness claim accordingly (the upper bound and all applications remain unaffected).
minor comments (4)
  1. [Proof of Theorem 1.1 (upper bound)] In the display after (14) the bound on h_L(a_i) for i≠m₁ uses m(k,k) in one place; this appears to be a typographical slip for m(k,j) or the already-chosen m₁/m₂.
  2. [Table 1] Table 1 caption says “current best known upper bounds /* for 2≤n≤8”; the n=2 entry is taken from [13] while the others are new—clarify this distinction in the caption or surrounding text.
  3. [Theorem 1.2 and its proof] The phrase “the resulting linear order /* is optimal” in Theorem 1.2 is justified only by the crude triangle-inequality lower bound 1+ε/n; a short remark that a matching linear lower bound with dimension-free constant remains open would be helpful.
  4. [References] Several arXiv identifiers in the bibliography carry future-looking dates (e.g., 2607.x); once the paper is revised these should be updated to the actual posted versions or replaced by journal references if available.

Circularity Check

1 steps flagged

Core upper bounds are self-contained first-principles convex geometry; only the tightness lower bound for n≥3 leans on authors’ unpublished/preprint planar and coning identities.

specific steps
  1. self citation load bearing [Section 3, Proposition 3.2 and Proof of the tightness in Theorem 1.1]
    "From relating these lengths also to the Minkowski asymmetry of the quadrangle, the following simple formula for trapezoids is obtained in [21]. Proposition 3.2. Let K∈K_2 be a trapezoid. Then d_BM(K,Δ_2)=3-s(K). … Propositions 3.3 and 3.4 show for any x∈R^n\aff(L×{0}) that K:=conv((L×{0})∪{x}) satisfies s(K)=s(L)+1=n-ε and d_BM(K,Δ_n)=d_BM(L,Δ_{n-1})=1+ε."

    The existence claim “there exists K with s(K)=n-ε and d_BM(K,Δ_n)=1+ε” (the “best possible up to linear order” half of Theorem 1.1) is proved by induction whose n=2 base case is imported verbatim from the authors’ unpublished working manuscript [21] and whose inductive step cites their concurrent preprint [22, Thm 1.4]. The lower-bound statement is therefore not re-derived inside the present paper; if the planar identity fails, the higher-dimensional existence claim collapses. This is load-bearing only for tightness, not for the upper-bound derivation.

full rationale

The load-bearing upper bound of Theorem 1.1 (Section 2) is derived from Klee/Böröczky contact-point structure, an explicit superset built from asymmetry-point simplices (Lemmas 2.4–2.11), and the Helly-type covering radius formula (Lemma 2.3); none of these steps define the claimed distance in terms of itself or fit a free parameter to the target. Theorems 1.2–1.3 likewise rest on a translated Loewner-ball inclusion (Lemma 4.1) proved via John decompositions and a variance inequality, answering Belloni–Freund without circular reduction. Self-citations ([9],[10],[14],[19]) supply independent background lemmas. The sole mild self-citation dependence is the existence half of Theorem 1.1 for n≥3, which inducts from a planar trapezoid identity taken from the authors’ unpublished manuscript [21] and a coning-distance preservation result from their concurrent preprint [22]. That affects only the matching lower-bound claim, not the upper-bound estimates or the applications (Theorems 1.2, 1.4). No equation equates a “prediction” to a fitted input by construction. Score 1 reflects this narrow, non-central dependence.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

Pure convex geometry: the paper rests on classical external theorems (John, Klee, Böröczky, Helly) and standard definitions (Minkowski asymmetry, Banach–Mazur distance, Loewner ellipsoid). No free parameters are fitted. The only non-standard named object is the ‘asymmetry point simplex’, which is a definition extracted from Klee/Böröczky contact points rather than a new physical entity.

axioms (5)
  • standard math John’s characterization of the Loewner ellipsoid (contact points and weights forming a decomposition of the identity)
    Invoked as Proposition 4.2 to prove Lemma 4.1.
  • standard math Klee’s uniqueness of the Minkowski center and existence of common boundary points when s(K)>n−1; Böröczky’s location of the center inside the asymmetry-point simplex
    Proposition 2.1 is the structural starting point of the entire Section 2 construction.
  • standard math Helly’s theorem implying the covering radius R(X,S) is attained on at most n+1 points
    Used in the proof of Lemma 2.3.
  • domain assumption s(K)≤n with equality iff K is a simplex; s(K)=1 iff K is centrally symmetric
    Classical Grünbaum facts used throughout to frame extremality.
  • ad hoc to paper Exact formula d_BM(trapezoid,Δ_2)=3−s(K) from the working manuscript [21]
    Load-bearing for the tightness induction in Section 3; not yet journal-published.
invented entities (1)
  • asymmetry point simplex independent evidence
    purpose: Named simplex formed by common boundary points of K and s(K)(c_K−K)+c_K; used as the approximating simplex throughout Section 2.
    Extracted from Klee/Böröczky contact geometry and given a convenient label; not a new geometric object beyond the classical contact points.

pith-pipeline@v1.2.0-daily-grok45 · 28528 in / 2963 out tokens · 65135 ms · 2026-07-30T12:55:09.530959+00:00 · methodology

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read the original abstract

We establish a tight stability estimate for the Minkowski asymmetry near its maximal value, improving previously known estimates due to B\"or\"oczky, Guo, and Schneider. More precisely, if an $n$-dimensional convex body $K$ has Minkowski asymmetry $s(K) \geq n-\varepsilon$ for $\varepsilon \in [0,1)$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + \varepsilon + \frac{\varepsilon^2}{2(1-\varepsilon)}. \] This dimension-independent estimate is best possible up to the linear order in $\varepsilon$. As an application, we prove a stability result for the maximal Banach-Mazur distance to the Euclidean ball, improving a previous estimate of Kobos to the optimal linear order: if $K$ is at Banach-Mazur distance at least $n - \varepsilon$ to the ball for $\varepsilon \in [0,\frac{1}{2})$, then its Banach-Mazur distance to the $n$-simplex is at most \[ 1 + 2\varepsilon + \frac{2\varepsilon^2}{1-2\varepsilon}. \] A key ingredient is a positive answer to a conjecture by Belloni and Freund, showing that every convex body $K$ contains a translated copy of its volume-minimal circumscribed ellipsoid scaled by a factor $1/\sqrt{n s(K)}$. Finally, we apply the stability estimate for the Minkowski asymmetry to obtain improved upper bounds for the diameter of the Banach-Mazur compactum in fixed dimensions.

discussion (0)

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Reference graph

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