An orthogonality inequality in $\mathbb{RP}^{n-1}$ and the double-cap conjecture

The orthogonal neighborhood of a set in real projective space is the union of the hyperplanes perpendicular to its points. We prove that caps minimize neighborhood measure among measurable sets of fixed measure in $\mathbb{RP}^{n-1}$, for $n \geq 3$, with rigidity up to null sets. This yields the do