Weak Simplex Conjecture

For every integer $n\ge 1$ , energy $E>0$ , and noise standard deviation $\sigma>0$ , let $x_0,\ldots,x_n\in\mathbb{R}^n$ satisfy $\lVert x_i\rVert^2=E$ for all $i$ , and let the signals have equal prior probabilities and be transmitted over an additive white Gaussian noise channel with maximum-likelihood decoding. If $P_c(x_0,\ldots,x_n;\sigma)$ denotes the probability of correct decoding, then $P_c$ is maximized by the vertices of a regular simplex centered at the origin, characterized by $\langle x_i,x_j\rangle=-E/n$ for all $i\ne j$ . Moreover, equality holds only when the signal set is a regular simplex, up to a permutation of the vertices and an orthogonal transformation of $\mathbb{R}^n$ .

Progress summary

Solved

A new preprint claims to have proved that the regular simplex is the unique best arrangement, but independent mathematical verification is not yet available.

The conjecture asserts that, for $n+1$ signals in $n$ -dimensional Euclidean space under equal energy limits, the regular simplex uniquely minimizes Gaussian decoding error. Its history runs from Shannon’s suspicion through the formulations of Cover (1987) and Massey (1988); the stronger version was disproved by Steiner (1994).

Known results

Balakrishnan gave an early partial study.
Dunbridge proved low-noise asymptotic optimality and local optimality at every noise level.
Landau and Slepian announced a proof in 1966, later found valid only for $n=3$ .
Steiner disproved the Strong Simplex Conjecture in 1994, leaving the weak version unresolved.

August 2026 equality-case claim

The preprint posted on August 19, 2026 claims to classify all equality cases, proving uniqueness of the regular simplex, including under an energy budget. It also reports a Lean 4 machine-check of the single-parameter rigidity argument. This is a claimed advance rather than independently verified resolution; a separate preprint had proved optimal values but explicitly left uniqueness open.

Current status (as of August 2026): the optimal-value statement has a preprint proof, while uniqueness is claimed by the August 2026 preprint but remains unverified.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

The Equality Cases of the Weak Simplex Conjecture — arXiv

1 day ago