  We are interested in the optimization of convex domains under a PDE
constraint. Due to the difficulties of approximating convex domains in
$\mathbb{R}^3$, the restriction to rotationally symmetric domains is used to
reduce shape optimization problems to a two-dimensional setting. For the
optimization of an eigenvalue arising in a problem of optimal insulation, the
existence of an optimal domain is proven. An algorithm is proposed that can be
applied to general shape optimization problems under the geometric constraints
of convexity and rotational symmetry. The approximated optimal domains for the
eigenvalue problem in optimal insulation are discussed.
