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 $9/(8\pi )$. 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.