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.
released 2026-09-24 | 4 theorems · 8 lemmas · 11 proofs · 8,379 words |
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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.