Propeller Conjecture

The propeller conjecture asserts that three planar sectors of angle maximize the sum of squared Gaussian first moments over measurable partitions, after extension in orthogonal directions. Precisely, if is the standard Gaussian measure on , the proposed bound for a set partition into measurable sets is

The first moments are unnormalized vectors, and the norms are Euclidean. Cell probabilities are unrestricted, and empty cells are allowed. For and , the three sectors, multiplied by their orthogonal complement, attain , with any remaining cells empty.

Heilman (2026) reported a proof of the bound in every dimension, with equality only for the three-sector construction, up to rotations, relabeling, and changes on sets of measure zero. He also gave a proof for dimension three that does not require computer assistance. The general-dimensional argument still uses numerical checks without rigorous interval error bounds, so the reported resolution requires further verification. Heilman (2026) builds on an AI-generated proof and develops a revised argument.