  In this paper we show how the method of parallel coordinates can be extended
to three dimensions. As an application, we prove the conjecture of Antunes,
Freitas and Krej\v{c}i\v{r}\'ik \cite{AFK} that "the ball maximises the first
Robin eigenvalue with negative boundary parameter among all convex domains of
equal surface area" under the weaker restriction that the boundary of the
domain is diffeomorphic to the sphere and convex or axiconvex. We also provide
partial results in arbitrary dimensions.
