  We prove that there exists a bounded convex domain $\Omega \subset
\mathbf{R}^3$ of fixed volume that minimizes the first positive curl eigenvalue
among all other bounded convex domains of the same volume. We show that this
optimal domain cannot be analytic, and that it cannot be stably convex if it is
sufficiently smooth (e.g., of class $C^{1,1}$). Existence results for uniformly
H\"older optimal domains in a box (that is, contained in a fixed bounded domain
$D \subset \mathbf{R}^3$) are also presented.
