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The Gaussian propeller conjecture in every dimension

Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most

9/(8\pi)

. In dimension at least two, three planar sectors of angle

2\pi/3

, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor

(8\pi/9)(1-1/k)

for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.

Manuscripts in this family · 1

The Gaussian propeller bound in every dimension

Sep 24, 2026·main.pdfLean

We prove the Gaussian propeller conjecture: for every finite measurable partition of a Euclidean space, the sum of the squared lengths of its Gaussian first moments is at most

9/(8\pi)

. For dimension at least two and at least three cells, three planar sectors of angle

2\pi/3

, extended by an orthogonal Euclidean factor, attain the bound.

Full abstract

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