The Gaussian propeller conjecture in every dimension
OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
Proves that the sum of squared Gaussian first moments of any finite measurable partition is at most 9/(8π). In dimension at least two, three planar sectors of angle 2π/3, extended orthogonally, attain the bound. Combined with the separate Unique Games theorem, this proves NP-hardness of improving the loss factor (8π/9)(1-1/k) for identity-target kernel clustering with fixed k ≥ 3 on rational centered positive semidefinite inputs.
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MANUSCRIPT 01
The Gaussian propeller bound in every dimension
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/(8π). For dimension at least two and at least three cells, three planar sectors of angle 2π/3, extended by an orthogonal Euclidean factor, attain the…
Full source abstract
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/(8π). For dimension at least two and at least three cells, three planar sectors of angle 2π/3, extended by an orthogonal Euclidean factor, attain the bound.
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