MathDB

The Standard Simplex Conjecture for Gaussian noise stability

Let n≥2n\geq2, let ρ∈[−1,1]\rho\in[-1,1], and let 3≤k≤n+13\leq k\leq n+1. Let {Ai}i=1k\{A_i\}_{i=1}^k be a measurable partition of Rn\mathbb{R}^n, meaning that ⋃i=1kAi=Rn\bigcup_{i=1}^k A_i=\mathbb{R}^n and γn(Ai∩Aj)=0\gamma_n(A_i\cap A_j)=0 for i≠ji\ne j. For bounded measurable f ⁣:Rn→Rf\colon\mathbb{R}^n\to\mathbb{R}, define

Tρf(x)=∫Rnf(xρ+y1−ρ2) dγn(y),T_\rho f(x)=\int_{\mathbb{R}^n}f(x\rho+y\sqrt{1-\rho^2})\,d\gamma_n(y),

where γn\gamma_n is standard Gaussian measure. A partition is regular simplicial conical if its parts are the Voronoi regions of the vertices of a regular simplex centered at the origin. The Standard Simplex Conjecture. If ρ∈(0,1]\rho\in(0,1] and γn(Ai)=1/k\gamma_n(A_i)=1/k for every ii, then

J=∑i=1k∫Rn1Ai(x)Tρ(1Ai)(x) dγn(x)J=\sum_{i=1}^k\int_{\mathbb{R}^n}1_{A_i}(x)T_\rho(1_{A_i})(x)\,d\gamma_n(x)

is maximized by a regular simplicial conical partition. If ρ∈[−1,0)\rho\in[-1,0), with no restriction on the measures of the parts, then JJ is minimized by a regular simplicial conical partition. This conjecture describes the optimal Gaussian rounding scheme for MAX-kk-CUT and is known in the two-part case; the source states that it remains open for k≥4k\geq4 and for negative or sufficiently large positive ρ\rho.

Sources & referencesView supporting material

Primary source

Steven Heilman, “Euclidean Partitions Optimizing Noise Stability”, arXiv:1211.7138 (2014).

Additional references

2 papers in this index state this conjecture (2009–2012). The statement above is taken from the most recent of them; the others are arXiv:0903.3362.

gaussian-geometrymax-k-cutnoise-stability

Progress summary

Refreshed
Partially solved

The conjecture has been proved only in limited cases, while its full equal-measure and negative-noise forms remain open.

The conjecture asserts that regular simplex partitions optimize Gaussian noise stability for equal parts under positive correlation, and minimize it without measure restrictions under negative correlation. The equal-measure problem is explicitly distinguished from the false unequal-measure extension.

Known results

  • The two-part case is known.
  • Heilman, Mossel, and Neeman proved the equal-measure case for k=3k=3, n≥2n\geq2, and sufficiently small positive ρ\rho.
  • Heilman established validity in certain bounded-dimensional regimes, with the dimension bound depending on the noise parameter.
  • De, Mossel, and Neeman (2017) proved computable approximate low-dimensionality, not exact optimality.

Unequal-measure counterexamples (2014)

Heilman, Mossel, and Neeman showed that shifted flat partitions defeat the standard simplex for prescribed unequal measures, for every k≥3k\geq3 and ρ≠0\rho\neq0. This does not contradict the equal-measure conjecture posed here.

Current status (as of August 2026): The two-part case and several restricted positive-noise regimes are settled, but the conjecture remains open for general k≥3k\geq3, including k≥4k\geq4, negative ρ\rho, and sufficiently large positive ρ\rho.

Sources

Solutions 0

No solutions have been posted yet.