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Stochastic Domination of Gaussian Maxima by the Regular Simplex

Abhijeet Mulgund

arXiv:2609.28452v2math.PRcs.ITmath.ITmath.MG

TL;DR

The paper addresses the conjecture that the regular simplex circumscribed about a centered ball minimizes Gaussian measure among simplices containing it. It proves an all-threshold comparison of Gaussian maxima, resolving the conjecture and yielding a unique signal geometry under the stated dimension and false-alarm conditions.

  • Problem

    Balitskiy, Karasev, and Tsigler conjectured that among simplices containing a centered ball, the circumscribed regular simplex minimizes Gaussian measure.

    Source: §1 · p. 2

  • Method

    The paper proves a stochastic comparison for maxima of centered Gaussian vectors with unit variances against the regular-simplex maximum.

    Source: §1 · p. 2, §1.1 · p. 3

  • Results

    For every threshold, the Gaussian maximum’s distribution function is at least the regular-simplex maximum’s, with equality at each fixed t>0 only for the simplex covariance.

    Source: §1.1 · p. 3

  • Takeaways & Limitations

    The comparison resolves the circumscribed-simplex conjecture and implies unique optimality of the regular simplex for signal identification at every positive false-alarm level when dimension is at least n-1.

    Source: §1 · p. 2, §1.1 · p. 3

    The paper leaves open the Gaussian-measure minimization problem for unit directions when the dimension is below n-1.

    Source: §7 · p. 24

Abstract

from arXiv · show

Let $n\ge2$, and let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector with $\mathrm{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standard Gaussians, and put $\overline{Z}=(Z_1+\cdots+Z_n)/n$. We prove $\mathbb{P}\{\max_i X_i\le t\}\ge\mathbb{P}\{\sqrt{n/(n-1)}\,\max_i(Z_i-\overline{Z})\le t\}$ for every $t\in\mathbb{R}$, and for each fixed $t>0$ equality holds only when $\mathrm{Cov}(X_i,X_j)=-1/(n-1)$ for all $i\ne j$. The right side is the distribution function of the maximum of the regular simplex vector. Equivalently, among all simplices containing a given centered ball, the regular simplex circumscribed about the ball has the least standard Gaussian measure, as conjectured by Balitskiy, Karasev, and Tsigler. In our preceding paper we proved this comparison after both maxima are smoothed by independent Gaussian noise of variance $1/(n-1)$, which suffices for the Weak Simplex Conjecture; here we remove the smoothing, which is what probabilities at a single threshold require. As an application we consider $n$ equally likely signals of equal energy in Gaussian noise, where the transmitter may also send nothing. At every positive false-alarm level, and for every law of a common nonnegative random amplitude not concentrated at zero, the regular simplex uniquely maximizes the average probability of correct identification whenever the signal dimension is at least $n-1$. A Lean formalization is available at https://github.com/abhmul/full-simplex-conjecture-lean.

1 Introduction

The paper proves an all-threshold stochastic comparison between unit-variance Gaussian maxima and the regular-simplex maximum, with a positive-threshold equality characterization. This also resolves the conjecture that the circumscribed regular simplex minimizes Gaussian measure among simplices containing a centered ball.

1.1 The main result

  • For every threshold, any unit-variance Gaussian maximum is stochastically dominated by the regular-simplex maximum, with equality at t>0 only for the simplex covariance.The equality condition is that all off-diagonal covariances equal -1/(n-1).Source: §1 · p. 2, §1.1 · p. 3
  • Among simplices containing a centered ball, the circumscribed regular simplex minimizes Gaussian measure, with equality only for its orthogonal images.The Gaussian-measure comparison is equivalent to the maximum comparison, and the paper proves the conjecture in every dimension.Source: §1 · p. 2, §1.1 · p. 3

1.2 Relation to the smoothed comparison

  • The preceding smoothed comparison does not by itself establish an unsmoothed threshold comparison, which the paper proves here.Smoothing suffices for exponential moments, but not for probabilities at a single threshold.Source: §1.2 · p. 4

1.3 The difficulty and the idea

  • Slepian’s inequality handles covariances entrywise above the simplex covariance, but other matrices require accounting for compensating correlations.Some pairs can be more negatively correlated than in the simplex while other pairs are less negatively correlated.Source: §1.3 · p. 4
  • At a covariance minimizing the maximum’s distribution function, individual conditional bounds may fail, but their aggregate yields the derivative inequality and its equality case.The paper reduces the theorem to this aggregate bound at a minimizer.Source: §1.3 · pp. 5–6

1.4 Consequences and applications

  • With an inactive signal state, the regular simplex uniquely maximizes average correct identification at every positive false-alarm level for nonzero nonnegative amplitude laws, when dimension is at least n-1.The receiver’s threshold depends on the upper α-quantile of the Gaussian maximum.Source: §1.4 · p. 6
  • The paper also proves an independence-ratio lower bound for arc-transitive graphs of every degree from the smoothed comparison.The underlying local-maximum probability bound holds at every eigenvalue, though only the least eigenvalue matters for independent sets.Source: §1.4 · p. 6

1.5 Organization

  • The proof proceeds from minimizers through threshold derivatives and first-order conditions to an aggregate bound; the author says the complete proof may be framed better.The paper expects to revise the exposition in later versions.Source: §1.5 · pp. 6–7

2 Reduction to a derivative bound at a minimizer

The proof reduces the Gaussian-maxima comparison to a derivative bound at a minimizing covariance, then uses induction to establish the inequality and its equality case.

2 Reduction to a derivative bound at a minimizer

  • The induction proves the comparison for dimension n using only the inequality in dimension n − 1 and a derivative bound for minimizing covariances.The argument first reduces the theorem to Proposition 2.2.Source: §2 · pp. 7–9
  • Compactness of the covariance set and continuity of the distribution function ensure that a minimizing covariance exists.The passages establish compactness and continuity, then use them to attain the minimum.Source: §2 · p. 7
  • At every positive threshold, equality in the comparison forces the minimizing covariance to be the regular simplex covariance.The equality clause applies at each fixed t > 0.Source: §2 · p. 9

3 Derivatives at the threshold

The section conditions on Gaussian coordinates to express threshold derivatives through conditional masses and pair weights, then establishes the regularity needed to use those derivatives.

3.2 Differentiability

  • The pair weights also enter the threshold derivatives, while singular covariance and unbounded regions make differentiability require a direct argument.The proof develops differentiability for the offset mass and then applies it to the threshold probability.Source: §3 · p. 9, §3.2 · pp. 10–11

3.1 Conditioning and the pair weights

  • Conditioning on X_i = t expresses the remaining constraints as a Gaussian region with offsets β_i, whose conditional mass is H_i.Pair weights q_ij use the density at X_i = X_j = t multiplied by the remaining conditional probability.Source: §3.1 · pp. 9–10

3.2 Differentiability

  • At β_i, the conditional mass is differentiable even though equal offsets can fail to be a differentiability point.The proof therefore differentiates at β_i when conditional hyperplanes may coincide at equal offsets.Source: §3.2 · p. 11

4 First-order conditions at a minimizing covariance

The section perturbs a minimizing covariance to derive matrix and row constraints that feed the aggregate bound.

4 First-order conditions at a minimizing covariance

  • At a minimizing covariance, the pair-weight matrix S is positive semidefinite and satisfies SG = 0.These conditions follow from the covariance perturbation argument.Source: §4 · pp. 12, 14
  • The passage gives κ > 0 and a_i > 0; it also states 0 < a_i < 1/2.Source: §4 · pp. 13–14
  • The kernel relation has a geometric consequence: the polyhedron defined by the minimizing covariance's normals and offsets is bounded.The passage links the positive coefficients k_i and the spanning normals to boundedness.Source: §4 · pp. 14–15

5 The aggregate bound and its equality case

The section combines conditional-mass inequalities into an aggregate bound and shows that equality identifies the regular simplex covariance.

5.1 Padding and two derivatives

  • Padding the conditional regions to dimension n − 2 preserves their masses and offset derivatives, allowing the row estimate to be used in the proof.The rank condition rank G ≤ n − 1 permits this padding.Source: §5.1 · p. 15

5.2 The reciprocal tangent and the dilation identity

  • Log-concavity of conditional masses yields a reciprocal tangent inequality, and dilation gives a positive decomposition H_i = u_i + e_i.The proof uses the two facts together in the aggregation step.Source: §5.2 · p. 16

5.3 From row inequalities to an aggregate bound

  • The aggregate bound gives Σ_i H_i ≥ n p_0 under the stated covariance and row conditions.The proof applies its scalar aggregation lemma with x_i = H_i, M = n p_0, and equal weights.Source: §5 · p. 15, §5.3 · p. 17

5.4 Equality

  • Equality in the aggregate bound forces every off-diagonal covariance entry to equal −1/(n − 1), so G is the regular simplex covariance.The equality argument combines proportionality of the relevant lists with positive semidefiniteness.Source: §5.4 · p. 18

5.5 Proof of the derivative bound

  • The aggregate bound supplies the derivative bound and its equality case needed to complete the proof of the main theorem.The proof applies Proposition 5.1 to conditional masses associated with a minimizing covariance.Source: §5.5 · p. 18

6 Consequences

The consequences extend the maximum comparison to integrated quantities and amplitude mixtures, identify the regular simplex as uniquely optimal for signal detection with an inactive state, and derive an independent-set bound for arc-transitive graphs.

6.1 Integrated comparisons

  • The maximum comparison implies Eψ(M_G) ≤ Eψ(M_∆n) for nondecreasing ψ, with strict inequality when G ≠ ∆n and ψ increases between some positive x < y.Positive independent scale mixtures preserve the comparison, strictly for t > 0 when G ≠ ∆n.Source: §6.1 · pp. 18–19
  • The integrated comparisons recover the regular simplex as the unique maximizer of mean width among convex hulls of n unit vectors in R^m when m ≥ n −1.Taking ψ(x) = e^λx also yields the Weak Simplex Conjecture for maximum-likelihood decoding, with uniqueness when m ≥ n −1.Source: §6.1 · pp. 19–20

6.2 Identification with an inactive state

  • For equally likely signals with an inactive state, the regular simplex uniquely maximizes average correct identification at every positive false-alarm level for common amplitude laws not concentrated at zero, when m ≥ n −1.The simplex also uniquely maximizes the smallest label-specific success probability in this dimension range.Source: §6.2 · pp. 21–22

6.3 Independent sets in arc-transitive graphs

  • For every nonempty finite d-regular arc-transitive graph, Corollary 6.3 gives an independent-set lower bound using its least adjacency eigenvalue, without the spherical-cap conjecture.The theorem-level comparison holds at every eigenvalue, although it does not improve the corollary's bound.Source: §6.3 · p. 23

7 Open problems

The section leaves open whether regular-simplex centers maximize the measure of equal spherical-cap unions at every radius, and asks which configurations minimize Gaussian measure when the dimension is too small for a regular simplex.

Spherical cap unions

  • For n ≥ 5, it remains open whether regular-simplex centers maximize the measure of the union of n equal spherical caps for every cap radius.An affirmative answer would imply the maximum comparison for correlation matrices of rank at most n −1, and hence for all such matrices.Source: §7 · p. 24
  • The cap-union question is resolved for n ≤ 4, while the Fejes Tóth moment argument does not apply when three caps of the regular configuration overlap.For n ≥ 5, such triple overlap occurs when the cap parameter is close to 1/(n −1).Source: §7 · p. 24

Low-dimensional configurations

  • When d ≥ 3 and n ≥ d + 2, the minimizing directions for the Gaussian-measure problem are unknown except for d = 3 with n = 6 or 12.For n ≤ d + 1, the minimum is attained only by the regular simplex; the cases d ≤ 2 are also resolved.Source: §7 · p. 24

A Equivalence with the circumscribed-simplex conjecture

The appendix proves that the non-strict circumscribed-simplex statement is equivalent to the Gaussian-maximum comparison, reducing general correlation matrices to simplex cases.

Equality case

  • In the equality case for a circumscribed simplex, the comparison forces the facet normals to form a regular simplex and every facet offset to equal t.If an offset exceeded t, a neighborhood of a point satisfying the defining inequalities would contradict the equality of Gaussian measures.Source: A Equivalence with the… · p. 25

Equivalence

  • The non-strict claim that the regular simplex circumscribed about B(0, t) minimizes Gaussian measure for every t > 0 is equivalent to the maximum-distribution comparison.The reduction follows the argument sketched in [BKT17, p. 4].Source: A Equivalence with the… · pp. 25, 27

Reduction

  • For a correlation matrix G other than J, the reduction lowers correlations entrywise to a matrix Γ with a nonnegative kernel vector, then approximates Γ by rank-(n−1) correlation matrices.The approximants have kernels spanned by vectors with strictly positive entries.Source: A Equivalence with the… · pp. 25–26

B Further identification criteria

This appendix section retains the notation introduced for the inactive-state identification problem.

Notation

B.1 Adversarial amplitudes

For a set of amplitude laws with a stochastically least law assigning positive probability to nonzero amplitudes, the simplex rule attains the optimal worst-case success value, and the regular simplex is the unique maximizing geometry. The positive-amplitude condition is essential when laws can approach zero.

B.1 Adversarial amplitudes

  • The simplex rule attains the adversarial-amplitude value, and the regular simplex is the only maximizing Gram matrix.The law set must contain a stochastically least law µ0 with µ0((0, ∞)) > 0.Source: §B.1 · p. 27
  • For each rule with false-alarm probability at most α, the upper-bound argument evaluates it at the least law µ0.The strict inequality for non-simplex Gram matrices follows from Theorem 6.2.Source: §B.1 · p. 27
  • The simplex rule succeeds with at least the value at µ0 for every law in the set and every label.The conditional success events increase with amplitude, and symmetry gives equal success probabilities across labels.Source: §B.1 · p. 27
  • If deterministic amplitudes in the law set approach zero, every rule has smallest success probability at most α/n.Uniformly outputting a random label with probability α and otherwise outputting 0 attains this bound for every geometry.Source: §B.1 · p. 27

B.2 Bayes accuracy

With an inactive state, the regular simplex uniquely maximizes Bayes accuracy when the amplitude law assigns positive probability to nonzero amplitudes.

B.2 Bayes accuracy

  • The optimal receiver’s accuracy is the expectation of the larger of the inactivity-prior term and the active-signal likelihood term.The maximizing-term decision rule attains this value, and the likelihood function ℓµ is increasing.Source: §B.2 · p. 28
  • For µ((0, ∞)) > 0 and every inactivity prior π0 ∈ [0, 1), the regular simplex uniquely maximizes Bayes accuracy.Equality holds only for Gram matrix G = ∆n.Source: §B.2 · p. 28
  • Strict increase of ℓµ beyond a threshold makes the comparison strict for every non-simplex geometry.The argument uses positive probability on an interval of positive amplitudes and π > 0.Source: §B.2 · p. 28

C Local maxima at every eigenvalue

The appendix extends the regular-simplex comparison to local-maximum probabilities at every adjacency eigenvalue of a finite arc-transitive regular graph. This yields the stated eigenvalue bound, including endpoint cases.

C Local maxima at every eigenvalue

  • The strict local maxima of the normalized Gaussian eigenvector form an independent set, linking the local-maximum probability to graph independence.The Gaussian vector is built from independent standard Gaussian coefficients on an eigenspace and rescaled to unit coordinate variances.Source: C Local maxima at every eigenvalue · p. 28
  • The proof converts the eigenvector’s neighbor values into a Gaussian comparison and applies Theorem 1.1 to obtain the bound.The intermediate smoothed comparison is extended from its restriction λ ≤ d − 2 to every eigenvalue.Source: C Local maxima at every eigenvalue · p. 29
  • At a fixed vertex, conditioning on its Gaussian value expresses strict local maximality as all neighbor values falling below a common threshold.Arc-transitivity makes the neighbor covariances equal, yielding a common threshold; this is its only use beyond vertex-transitivity.Source: C Local maxima at every eigenvalue · p. 29
  • For every adjacency eigenvalue λ of a finite simple d-regular arc-transitive graph with d ≥ 3, pG(λ) ≥ qd(λ).At λ = −d both probabilities are 1/2; at λ = d both are zero.Source: C Local maxima at every eigenvalue · p. 29
  • For degree 3, the proposition is an identity; for other arc-transitive graphs of degree d ≥ 5, the result is new only beyond the λ ≤ d−2 case, which already follows from (1.7).For d = 4, the result follows from Fejes Tóth’s theorem; for cherry-transitive graphs, equality holds in every degree [HV15, Definition 2.7, Proposition 2.8, and Definition 2.9].Source: C Local maxima at every eigenvalue · p. 30

D AI disclosure

The disclosure describes a research system and the author’s role in directing its work toward the simplex conjecture. It also reports model contributions to proof development, writing, and Lean formalization, alongside independent checking and author responsibility.

D AI disclosure

  • The system organized more than two hundred Gaussian extremal problems and alternated parallel attacks with compression of results into a smaller foundation.It was designed to develop theory across related problems and deepen the author’s understanding of Gaussian extremal problems.Source: D AI disclosure · p. 30
  • The author made the all-correlation-matrices statement the primary target, and the proof for all n followed a compressed proof strategy developed from the n = 5 case.The reported progression includes proofs for n = 3 and n = 4, then n = 5 and all dimensions.Source: D AI disclosure · pp. 30–31
  • The author brought compression phases from earlier work and steered the system, while exchanges with the coordinator shaped two organizing proof ideas.The joint minimization over covariance and threshold began the n = 5 proof and became part of the proof of Theorem 1.1.Source: D AI disclosure · p. 31
  • Models produced the technical proof steps, while the author independently checked the mathematical arguments and references and retained responsibility for correctness and attribution.The disclosure says no theorem, proof step, or bibliographic assertion was kept solely on automated authority.Source: D AI disclosure · p. 32
  • A Codex agent formalized Theorem 1.1 in Lean, and the Lean kernel checker accepts the formalization.Independent agents reviewed it; the author plans to release the research system after clearing personal and sensitive information.Source: D AI disclosure · p. 32
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