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A simplex ratio recovers the Minkowski measure of symmetry, characterises simplices by outer additivity, and proves low-depth polytopes cannot approximate them.

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2026-07-11 23:45 UTC pith:HCWUQAHY

load-bearing objection Clean convex-geometry paper: new simplex ratio, sharp outer-additivity characterization, improved stability, and a sharp depth obstruction for approximating simplices that is independent of the Kadets input.

arxiv 2607.03815 v1 pith:HCWUQAHY submitted 2026-07-04 math.MG cs.LG

A simplex-based measure of symmetry

Egor Bakaev , Amir Yehudayoff This is my paper
classification math.MG cs.LG MSC 52A2052A4052B11
keywords measure of symmetryMinkowski measuresimplexBanach-Mazur distanceouter additivitydepth complexityinput convex neural networkspolytopes
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 introduces a simple ratio that compares the smallest positive and negative simplex homothets needed to cover a convex body. Taking the supremum of this ratio over all centered simplices recovers the classical Minkowski measure of symmetry. With that identification the authors improve the known stability theorem: any body whose Minkowski measure is within ε of the maximum n is at most 1/(1-ε) away from a simplex in Banach–Mazur distance. They also prove that the only convex bodies for which the outer-containment size is additive under Minkowski summation are the simplices themselves. Finally, for polytopes assembled by nested Minkowski sums and convex hulls of unions—the operations that define depth for input-convex ReLU networks—they obtain the sharp bound that the simplex ratio is at most 2^d-1 at depth d, so simplices cannot be approximated by shallow constructions.

Core claim

The simplex-based measure ρ_Δ(L)=λ_-Δ(L)/λ_Δ(L) has an affine-invariant envelope equal to the Minkowski measure of symmetry. The same quantity is controlled by the two operations that generate depth-d polytopes (Minkowski sum never increases it beyond the worst summand; convex hull of a union at most adds the two values plus one), yielding the sharp upper bound ρ_Δ(P)≤2^d-1 whenever the depth of P is d. Consequently any polytope of depth less than log_{2}(n+1) stays a definite distance from the simplex.

What carries the argument

The ratio ρ_Δ of the two outer coefficients with respect to a fixed simplex Δ, expressed via the average of support-function values on the n+1 facet normals of a centered simplex. This single number is simultaneously an affine-invariant proxy for Minkowski symmetry, an additive invariant that characterises simplices, and a depth-monotone quantity for polytopes.

Load-bearing premise

The upper comparison between the simplex ratio and the inner-to-outer homothety ratio rests on reducing to a smooth strictly convex body and then applying a Kadets-type theorem for inductive coverings of space by supporting half-spaces.

What would settle it

Produce either a non-simplex that is outer additive, or a depth-d polytope in dimension n with d<ceil(log_{2}(n+1)) whose simplex ratio exceeds 2^d-1, or a body with Minkowski measure at least n-ε whose Banach–Mazur distance to the nearest simplex is strictly larger than 1/(1-ε).

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

If this is right

  • Any convex body whose Minkowski measure is at least n-ε lies at Banach–Mazur distance at most 1/(1-ε) from a simplex.
  • A convex body makes outer containment size additive under Minkowski sum if and only if it is a simplex.
  • Every polytope of depth complexity d satisfies ρ_Δ≤2^d-1, and the bound is attained by certain simplices of dimension 2^d-1.
  • Support functions of depth-d polytopes remain a definite L1 distance away from the max-of-(n+1)-linears function whenever d is smaller than log_{2}(n+1).
  • The same depth obstruction applies to every body of full Minkowski measure n, not only to the regular simplex.

Where Pith is reading between the lines

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

  • The same logarithmic-depth lower bound is likely to obstruct shallow approximation of any continuous piecewise-linear function that is complete for the max function, limiting the expressivity of shallow input-convex networks more broadly.
  • Outer additivity supplies a new axiomatic route to recognising simplices that may be useful when studying indecomposability or approximation by Minkowski sums.
  • The inductive-covering technique used for the upper bound on ρ_Δ can probably be reused to obtain quantitative stability statements for other classical measures of asymmetry.

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

0 major / 5 minor

Summary. The paper defines a simplex-based (non-affine-invariant) measure of symmetry ρ_Δ(L) := λ_{-Δ}(L)/λ_Δ(L) via smallest containing homothets of the regular simplex and its negative. It proves four main results: (1) the classical Minkowski measure m*(K) equals the supremum of ρ_Δ over all origin-barycentered simplices in the affine hull (Theorem 2); (2) an improved stability estimate: m*(L) ≥ n-ε implies Banach–Mazur distance at most 1/(1-ε) to a simplex (Theorem 5), via the comparison n/d_Δ(L) ≤ ρ_Δ(L) ≤ n-1 + 1/d_Δ(L) (Theorem 3); (3) a characterization that a convex body K is outer-additive (λ_K additive under Minkowski sum) if and only if K is a simplex (Theorem 6); (4) for the depth complexity of polytopes generated by Minkowski sums and convex hulls of unions, the sharp bound ρ_Δ(P) ≤ 2^d - 1 whenever depth(P) = d (Theorem 7), with the consequence that low-depth polytopes cannot approximate simplices and that m*(P) is likewise bounded (Theorem 8), plus an inapproximability transfer to support functions of input-convex ReLU networks.

Significance. The work cleanly unifies classical convex geometry (Minkowski measure, stability, indecomposability/characterizations of simplices) with a modern computational model of polytopes arising from ReLU/ICNN expressivity. The depth bound is sharp, elementary (induction via support-function averaging for sums and a max-bound for conv-union), and yields a uniform geometric obstruction to approximating the simplex (hence the max function) by low-depth constructions; this is a genuine contribution beyond asymptotic indecomposability results of Shephard type. The outer-additivity characterization is novel and self-contained. The stability improvement is asymptotically sharper than Schneider’s earlier bound and holds for the full range ε < 1. All arguments are written in full detail from support functions and Carathéodory; the only external input is the Akopyan–Karasev Kadets-type theorem, used solely for the upper half of Theorem 3 (hence only for stability, not for the depth results).

minor comments (5)
  1. [§4.2, Lemma 18] In the proof of Lemma 18 the reduction to smooth strictly convex bodies is asserted by density; a one-sentence justification that both ρ_Δ and d_Δ are continuous in the Hausdorff metric (or that the inequality passes to the limit) would make the argument fully self-contained.
  2. [§4.2, Claim 21] Claim 21 verifies inductiveness of the covering by a general J-covering argument; the base case and the two inductive steps are clear, but a short remark that the covering may have fewer than n+1 sets (k ¤ n+1) does not affect the application of Theorem 20 would help the reader.
  3. [§6.3, proof of Theorem 7] The sharpness construction for Theorem 7 (a (2^d-1)-dimensional simplex built from vertices of Δ) is only sketched; spelling out the inductive construction of the vertex set would make the equality case completely explicit.
  4. [throughout] Notation for the model simplex alternates between Δ, Δ^n and Δ_n; a single consistent choice (e.g., Δ_n throughout) would improve readability.
  5. [§1.1] In the introduction the phrase “measures of symmetry capture how close … and are (assumed to be) affine invariant” is slightly awkward; the parenthetical can be dropped or rephrased.

Circularity Check

0 steps flagged

No circularity: all four main results are derived from support-function identities and elementary properties of Minkowski sum / convex hull, without fitted parameters or load-bearing self-citations.

full rationale

The paper defines ρ_Δ(L) := λ_{-Δ}(L)/λ_Δ(L) via the classical support-function formula (Proposition 14) and then proves four independent statements. (1) α(K) = m^*(K) follows by applying Carathéodory to the set of maximizers of the Minkowski ratio (Lemma 16 + Corollary 15). (2) The improved stability bound is obtained from the comparison n/d_Δ ≤ ρ_Δ ≤ n-1 + 1/d_Δ (Theorem 3), whose upper half uses only an inductive covering of supporting half-spaces (Claim 21) and the external Akopyan–Karasev theorem. (3) Outer additivity characterises simplices by a local geometric obstruction (Lemma 24) plus induction on dimension. (4) The depth bound ρ_Δ(P) ≤ 2^d-1 is proved by induction on depth using only the two elementary inequalities for Minkowski sum and conv-of-union (Theorems 9–10). No quantity is fitted to data, no uniqueness theorem is imported from the authors’ prior work, and the few self-citations (BBH+25/26, Val24) appear only as background motivation for the depth model; they are never used inside any proof. The derivation chain is therefore self-contained and free of circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 3 invented entities

The paper works entirely inside standard convex geometry (support functions, homothets, Banach-Mazur distance, Caratheodory). The only external non-elementary theorem is Akopyan-Karasev on inductive coverings. No free parameters are fitted; the invented notions (rho_Delta, outer additivity, polytope depth) are defined explicitly and used as tools rather than postulated entities with independent physical existence.

axioms (4)
  • standard math Additivity of support functions: h_{L1+L2}=h_{L1}+h_{L2}
    Used throughout Sections 2 and 5 to relate lambda under Minkowski sum.
  • standard math Akopyan-Karasev Kadets-type theorem: sum of inner coefficients over an inductive covering is at least 1
    Invoked as Theorem 20 to prove the upper bound of Lemma 18 that feeds stability and depth results.
  • standard math Caratheodory theorem in Euclidean space of dimension k
    Used to extract finitely many extreme directions for Minkowski centers and supporting half-spaces.
  • domain assumption Density of smooth strictly convex bodies in the Hausdorff metric
    Invoked in the proof of Lemma 18 to reduce to the smooth case before constructing the inductive covering.
invented entities (3)
  • simplex-based measure rho_Delta no independent evidence
    purpose: Provides a computationally simple, non-affine-invariant ratio that becomes the Minkowski measure after affine maximization and controls depth complexity.
    Defined in Section 1.2; all subsequent theorems are statements about this ratio or its affine version.
  • outer additivity no independent evidence
    purpose: Isolates the functional property lambda_K(L1+L2)=lambda_K(L1)+lambda_K(L2) that characterises simplices.
    Introduced in Section 5; the characterisation theorem is the main result of that section.
  • depth complexity of polytopes under sum and conv-union no independent evidence
    purpose: Models the hidden-layer depth of input-convex ReLU networks via support functions.
    Defined in Section 6; the bound rho_Delta <= 2^d-1 is the central complexity statement.

pith-pipeline@v1.1.0-grok45 · 23503 in / 2588 out tokens · 25354 ms · 2026-07-11T23:45:25.365037+00:00 · methodology

0 comments
read the original abstract

For compact convex sets $L,K \subset \mathbb{R}^n$, denote by $\lambda_K(L)$ the smallest size of a homothet of $K$ that contains $L$. We define a measure of symmetry based on the $n$-simplex $\Delta = \Delta^n \subset \mathbb{R}^n$ as the ratio \[ \rho_\Delta(L):=\frac{\lambda_{-\Delta}(L)}{\lambda_{\Delta}(L)}. \] We study this measure and deduce the following results: (1) The classical Minkowski measure of symmetry $m^*(L)$ can be defined as an affine-invariant version of $\rho_\Delta(L)$. (2) We improve the stability analysis for the Minkowski measure of symmetry; if $m^*(L)\ge n-\varepsilon$ then $L$ is $\tfrac{1}{1-\varepsilon}$-close to $\Delta$ in the Banach--Mazur distance. (3) We obtain a novel characterization of simplices as the only convex bodies $K$ for which the function $L \mapsto \lambda_K(L)$ is additive (a property we term ``outer additivity''). (4) Motivated by the expressivity of ReLU neural networks, we study the depth complexity of polytopes in $\mathbb{R}^n$ under the two operations: Minkowski sum and convex hull of a union. We prove the sharp bound $\rho_\Delta(P) \leq 2^d -1$ for every polytope $P$ of depth complexity $d$. In other words, simplices cannot be approximated by low-depth polytopes.

Figures

Figures reproduced from arXiv: 2607.03815 by Amir Yehudayoff, Egor Bakaev.

Figure 1
Figure 1. Figure 1: An illustration of ρ∆(L). We write ∆ = ∆n for the standard regular simplex in R n obtained by orthogonally projecting conv{e1, . . . , en+1} ⊂ R n+1 to the subspace orthogonal to (1, 1, . . . , 1), where e1, . . . , en+1 are the standard unit vectors. Definition. For a compact convex set L ⊂ R n that is not a single point, the simplex￾based measure is ρ∆(L) := λ−∆(L) λ∆(L) . By convention, if L is a point,… view at source ↗
Figure 2
Figure 2. Figure 2: An illustration of dK(L). the simplex and m∗ (∆) = n. Stability refers to a statement of the form: if m∗ (L) is close to the maximum n then L is close to ∆ in the Banach–Mazur distance dBM. Stability estimates of this kind were obtained in [BJ96, BJ05, Guo05, Tot13], and to the best of our knowledge the strongest previously known result is due to Schneider (Theorem 2.1 in [Sch09], see also [Tot15], Theorem… view at source ↗
Figure 3
Figure 3. Figure 3: The “local” condition for being outer additive from Lemma 24. Setting α = min{α1, α2}, we conclude that the following parallelogram L lies in K: L = conv{ a+y 2 , a+y 2 + α(x−y) 2 , b+y 2 , b+y 2 + α(x−y) 2 }. It can be expressed as L = L1 + L2 with L1 = [ a+y 2 , b+y 2 ] and L2 = [0, α(x−y) 2 ]. Because L ⊂ K, λK(L) ≤ 1. The interval L1 is parallel to [a, b]. Because a ∈ Au and b ∈ Bu, λK([a, b]) = 1, so … view at source ↗

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

    math.MG 2026-07 accept novelty 6.0

    If s(K) ≥ n−ε then d_BM(K, simplex) ≤ 1+ε+ε²/(2(1−ε)), optimally linear in ε, with applications to ball-distance stability and Banach–Mazur diameter bounds.

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