  In the optimization of convex domains under a PDE constraint numerical
difficulties arise in the approximation of convex domains in $\mathbb{R}^3$.
Previous research used a restriction to rotationally symmetric domains to
reduce shape optimization problems to a two-dimensional setting. In the current
research, two approaches for the approximation in $\mathbb{R}^3$ are
considered. First, a notion of discrete convexity allows for a nearly convex
approximation with polyhedral domains. An alternative approach is based on the
recent observation that higher order finite elements can approximate convex
functions conformally. As a second approach these results are used to
approximate optimal convex domains with isoparametric convex domains. The
proposed algorithms were tested on shape optimization problems constrained by a
Poisson equation and both algorithms achieved similar results.
