The equal-radius simplex conjecture for Gaussian error probability

From papers

Let $P(v_1,\ldots,v_N)=\int_{\mathbb R^n}\max_i e^{-|x-v_i|^2}\,dx$ for a code consisting of $N$ vectors in $\mathbb R^n$ . The simplex conjecture. For $N=n+1$ and fixed $|v_1|=\dots=|v_N|=r$ , the maximum of $P(v_1,\ldots,v_N)$ is attained at any configuration forming a regular simplex inscribed into the ball of radius $r$ . The original fixed-total-energy version is false, but this equal-radius formulation is the conjecture studied in the paper and is proved only in dimensions at most $3$ .

Progress summary

Solved

A 2023 preprint claims a proof in every dimension, but independent verification is not recorded, so only the low-dimensional cases are settled.

The conjecture says that, with one more codeword than the dimension and equal signal energies, the regular simplex maximizes Gaussian decoding probability. Its origins are traced to Shannon; the equal-radius version is distinct from the disproved fixed-total-energy conjecture.

Known results

The equal-radius conjecture is established in dimensions through three; the three-dimensional argument does not extend directly because higher dimensions permit triple cap intersections.
Landau and Slepian, 1966, gave a purported general proof, but its geometric argument works only in dimension three.
Steiner, 1994, disproved the fixed-total-energy version for sufficiently many codewords; the regular simplex remains locally and asymptotically optimal there.

2023 claimed proof

A 2023 preprint claims to prove the Weak Simplex Conjecture in arbitrary dimension by jointly relaxing the codebook and decision regions, concluding that regular simplices are optimal. The claim is presented as a proof, but the retrieved record contains no referee verification, correction, withdrawal, or retraction.

Current status (as of August 2026): The equal-radius conjecture is settled through three dimensions, while the claimed arbitrary-dimensional proof remains unverified.

Sources

Sources & referencesView supporting material

Primary source

Alexey Balitskiy, Roman Karasev and Alexander Tsigler, “Optimality of codes with respect to error probability in Gaussian noise”, arXiv:1701.07986 (2017).

1 week ago

The current posted version of Pastore's A Proof of the Weak Simplex Conjecture (arXiv:2306.13478v2, 13 November 2023) does not establish its advertised all-dimensional theorem.

The injectivity argument in Section III-D defines conditional means $t,t'$ over the cone differences $\widetilde D\setminus\widetilde D'$ and $\widetilde D'\setminus\widetilde D$ in (18c)--(18d). The appendix then uses the set inclusions (23b)--(23c) as though those conditional means themselves belonged to the corresponding nonconvex set differences, yielding the sign statements (24b)--(24c). That inference is invalid: averaging preserves membership in a convex set, not in an arbitrary nonconvex set difference.

Appendix B.2 of Mulgund, arXiv:2607.14087v2, gives an explicit three-dimensional Gaussian simplicial-cone counterexample in which

\mathbb E[Z\mid Z\in D\setminus D']\in\operatorname{int}D',

so the required membership/sign inference actually fails in the setting of the proof. The same appendix gives an independent admissible example contradicting the facet-normal pairwise-independence assertion used to normalize $\lambda_1=1$ in the appendix of A Proof of the Weak Simplex Conjecture.

These defects occur in the injectivity result used by Section III-E to deduce the reflection symmetry needed for the final theorem. This challenge concerns only the posted v2 on arxiv; it does not assert that the conjecture is false or that the approach cannot be repaired.

Supporting discussion: Appendix B.2 of arXiv:2607.14087v2.

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Abhijeet Mulgund · 1 week ago · edited

Abhijeet Mulgund, Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture, arXiv:2607.14087v2, claims a complete resolution of this conjecture in every dimension.

The equivalence with the formulation on this page is immediate by scaling. Write $N=n+1$ , $v_i=r x_i$ with $\|x_i\|=1$ , and put $\lambda=\sqrt{2}\,r$ . If

P(v_1,\ldots,v_N) = \int_{\mathbb R^n}\max_i e^{-\|z-v_i\|^2}\,dz,

then the average probability of correct maximum-likelihood decoding in the Gaussian model of the paper satisfies

\psi_{\sqrt{2}r}(x_1,\ldots,x_N) = \frac{1}{N\pi^{n/2}}\,P(v_1,\ldots,v_N).

The multiplicative factor is independent of the configuration. Thus maximizing the MathDB objective $P$ over equal-radius codewords is exactly the Weak Simplex optimization solved in Corollary 2.4 of arXiv:2607.14087v2: a regular simplex is a maximizer for every radius $r>0$ .

Solution source: Abhijeet Mulgund, Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex Conjecture, arXiv:2607.14087v2 (2026).

Formal verification: The equivalent normalized Gaussian maximum-likelihood theorem is kernel-checked in Lean 4 at commit dcf1a45bc22ec54775314927b2a95fbe7c630edf; see WeakSimplex.weak_simplex and WeakSimplex.weak_simplex_of_scoreMaximizingDecoders. For positive signal strength, the formalization also proves equality if and only if the code Gram matrix is the regular-simplex Gram matrix (WeakSimplex.weak_simplex_eq_iff_codeGram_eq_of_scoreMaximizingDecoders). The scaling identity connecting this theorem to the displayed equal-radius integral is supplied above but is not itself a standalone Lean declaration.

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Abhijeet Mulgund · 1 week ago · edited