Why three Gaussian sectors can beat arbitrarily many cells

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Family 096 · Convex and metric geometry
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-14

1. The question

Divide Gaussian space into finitely many measurable cells. For each cell, add up its position vectors, weighted by Gaussian probability, and square the length of the resulting vector. How large can the sum of those squared lengths be?

One might expect that more dimensions or more cells would permit increasingly elaborate partitions with larger values. The propeller conjecture predicts a strikingly simple answer: once two dimensions and three labels are available, three equal planar sectors are already optimal. Extra dimensions are ignored, and extra cells can be left empty.

The manuscript in family 096 claims this bound for every dimension and every finite number of cells. It uses the established three-dimensional propeller theorem and supplies an argument excluding improvements by extremizers with five or more active cells. A separate application evaluates a threshold in kernel clustering, with an additional dependency for the hardness conclusion.

2. Background and essential definitions

The standard Gaussian probability measure on $\mathbb R^d$ has density

$$(2\pi)^{-d/2}e^{-|x|^2/2}.$$

For a measurable cell $A$, define its first moment

$$z(A)=\int_A x\,d\gamma_d(x).$$

This is not divided by $\gamma_d(A)$. When the cell has positive probability, it equals that probability times the conditional mean of a Gaussian vector in the cell. The source calls it an unnormalized centroid.

For a partition $A_1,\ldots,A_k$, the objective is

$$F(A_1,\ldots,A_k)=\sum_{i=1}^k|z(A_i)|^2.$$

The cells may have unequal probabilities, and empty cells are allowed. Partitions are considered up to Gaussian null sets, which do not affect the integrals. Since the whole Gaussian distribution has mean zero, the cell moment vectors sum to zero.

A propeller partition consists of three sectors of angle $2\pi/3$ in a plane. In higher dimensions, take each sector times the orthogonal complement of the plane. The additional coordinates contribute zero to each first moment by symmetry.

For a planar sector of opening $2\alpha$, symmetric about a unit direction $u$, polar integration gives

$$z(A)=\frac{\sin\alpha}{\sqrt{2\pi}}u.$$

Putting $\alpha=\pi/3$ and summing over three sectors gives $3\sin^2(\pi/3)/(2\pi)=9/(8\pi)$. The conjecture is that no other finite partition can exceed this value.

3. Development history and previous partial results

Khot and Naor introduced the propeller problem in their Approximate kernel clustering. Gaussian first moments enter both their rounding analysis and their hardness framework for a discrete clustering objective.

Their geometric reduction showed that an optimizer can be sought among conical partitions determined by linear scores. It also reduced the effective ambient dimension to the span of the cell moments. They determined the three-cell value, where the propeller already supplies the optimum.

Heilman, Jagannath and Naor proved the propeller conjecture in three dimensions in 2013. Their computer-assisted argument treated the remaining four-cell conical configurations through spherical geometry and finite estimates. The present manuscript explicitly uses that theorem as an established input.

Higher-dimensional partitions can have more active cells, so the three-dimensional result does not automatically settle the full conjecture. The new argument focuses on this remaining regime. Its other geometric input is Ehrhard’s Gaussian convex-set inequality, which controls Gaussian measures of Minkowski combinations and, in the form used here, translations of convex cells.

A related literature studies noise stability: the probability that two correlated Gaussian vectors fall in the same cell, often with equal cell probabilities prescribed. Heilman’s later work gives a conditional route to propeller bounds in restricted cell counts under additional stability assumptions. That objective and those mass constraints differ from the unrestricted first-moment problem here.

4. What the new manuscripts claim

M1 states that for all positive integers $d,k$, every measurable partition of $\mathbb R^d$ into $k$ cells satisfies

$$\sum_{i=1}^k\left|\int_{A_i}x\,d\gamma_d(x)\right|^2 \leq\frac9{8\pi}.$$

The constant is attained whenever $d\geq2$ and $k\geq3$, using the propeller and leaving additional cells empty. The theorem does not assert that the same constant is attained when there is only one dimension or fewer than three labels. It identifies the maximum value and an attaining construction in the stated range, without giving a full classification of all equality cases.

The separate clustering application takes a rational symmetric positive semidefinite matrix $A=(a_{pq})$ satisfying $A\mathbf1=0$. It seeks a labeling $\sigma$ of the $N$ indices by $k$ labels that maximizes

$$\sum_{p,q=1}^N a_{pq}\,\mathbf1_{\{\sigma(p)=\sigma(q)\}}.$$

For fixed $k\geq3$, define

$$\alpha_k=\frac{8\pi}{9}\left(1-\frac1k\right).$$

A loss factor $\alpha$ means achieving at least $1/\alpha$ of the optimum. M1 quotes Khot–Naor’s expected rounding guarantee at $\alpha_k$ in an ideal Gaussian-sampling and exact-relaxation formulation.

Using the newly evaluated Gaussian constant, their low-influence theorem, and the separate Unique Games Theorem in family 102, M1 claims NP-hardness of deterministic approximation with every fixed loss factor strictly below $\alpha_k$. The Unique Games dependency belongs to this hardness application, not to the Gaussian geometric inequality.

5. Why the advance matters

Importance: 4/5. The result would settle a prominent Gaussian partition problem whose low-dimensional case required substantial work. It reveals that permitting many cells and dimensions does not improve this particular objective beyond a planar three-part structure.

The geometry is tied to a concrete optimization threshold. Evaluating a continuous Gaussian extremum determines a constant in the analysis of discrete clustering algorithms and, with additional complexity inputs, their limitations. This is a recurring theme in approximation theory: a geometric inequality can encode an algorithmic barrier.

The number of active cells is part of the phenomenon. A formulation requiring every cell to have equal positive mass would prevent unused labels and would ask a different question. Here the option to concentrate the partition into three active sectors is essential to the proposed answer.

The rating reflects a major result with a precise cross-disciplinary consequence, while remaining narrower than a general classification of Gaussian partition optimizers or noise-stability extremizers. It does not treat the dependent hardness theorem as a consequence of geometry alone.

6. The main idea

Replace measurable sets by competing linear scores

Optimizing directly over all measurable partitions looks infinite-dimensional. Norm duality converts the square root of the largest objective into the largest expected maximum of finitely many Gaussian linear scores, with the score vectors’ total squared norm normalized to one.

For a fixed list of vectors, assigning a point to its largest score gives a conical partition. Compactness supplies an optimizer. Among optimizers, the argument chooses one with the fewest active cells. If two nonzero centroids had a nonnegative inner product, merging their cells would not decrease the objective, conflicting with that minimal choice unless the configuration could be simplified further.

The resulting centroids have strictly negative pairwise inner products and span a space of dimension one less than the number of active cells. Thus four or fewer active cells reduce to the established three-dimensional theorem. Any counterexample must have at least five.

Compare the cost of deleting one score

Normalize the centroid lengths to numbers $r_i$ with $\sum_i r_i^2=1$, and write $P_i$ for cell probabilities, with $\sum_iP_i=1$.

Deleting one score reduces the expected maximum. Optimality gives a lower bound on that loss after recentering the remaining score vectors. Ehrhard’s inequality, applied to translations of the deleted score’s winning cone, supplies an upper bound. Comparing them constrains how small a cell probability can be relative to its centroid length.

Halfspace rearrangement provides another one-cell estimate. In the five-cell case, an additional spherical comparison in the four-dimensional centroid span strengthens the constraint. These are restrictions on each cell separately, but they do not yet rule out the whole configuration.

Make two competitors constrain each other

The argument selects the cell with smallest centroid length and projects other scores orthogonally to its centroid direction. Gaussian orthogonality makes the residual scores independent of the component in that direction.

Their first-moment identities over the selected cell vanish. For a pair of residual scores, those identities yield a bound involving the determinant of their covariance matrix. The peak density of a two-dimensional Gaussian is inversely proportional to the square root of that determinant, linking geometric competition to a quantitative probability estimate.

The pairwise inequalities constrain the largest centroid lengths relative to the smallest one. Combined with the one-cell lower bounds and the fixed total probability mass, they become incompatible with an objective above $9/(8\pi)$. The manuscript supplies finite rational enclosures for the final scalar comparisons, in addition to the geometric reductions.

7. Scope, limitations, and open questions

The objective uses unnormalized first moments. Dividing each vector by the cell probability would change the optimization problem completely. The measure is the standard Gaussian, and the partition has finitely many measurable cells with no prescribed masses.

The theorem gives the universal bound and propeller attainment for $d\geq2,k\geq3$. It does not claim that every equality partition is a propeller, or that the propeller optimizes other Gaussian objectives such as equal-mass noise stability.

The clustering threshold separates an expected randomized rounding guarantee from deterministic hardness of a strict improvement. The manuscript does not supply an exact finite-bit implementation at the endpoint. Deterministic NP-hardness alone also does not exclude randomized algorithms under the sole assumption $P\ne NP$.

The geometric proof depends on the established three-dimensional theorem and Ehrhard’s inequality. The hardness application adds the companion Unique Games claim and a discrete low-influence theorem. These dependencies are reported as stated; no proof or computational certificate is checked in this digest.

8. Sources and further reading

M1’s introduction defines the objective, computes the propeller value and separates the geometry from clustering. Its middle sections develop score deletion and two-competitor estimates; its final application specifies the rational-input hardness statement.

The linked Khot–Naor paper is the primary source for the conjecture and rounding framework. Heilman–Jagannath–Naor supplies the three-dimensional theorem used as input, while Ehrhard’s paper gives the Gaussian convexity inequality behind the deletion estimate.

Source manuscripts

• M1. OpenAI, The Gaussian propeller bound in every dimension (September 24, 2026).

Sources consulted on 14 October 2026 at repository revision 1581d4c. Download the manuscript citations.