  We develop a numerical method for the computation of a minimal convex and
compact set, $\mathcal{B}\subset\mathbb{R}^N$, in the sense of mean width. This
minimisation is constrained by the requirement that
$\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$ for all unit vectors
$u\in S^{N-1}$ given some Lipschitz function $C$.
  This problem arises in the construction of environmental contours under the
assumption of convex failure sets. Environmental contours offer descriptions of
extreme environmental conditions commonly applied for reliability analysis in
the early design phase of marine structures. Usually, they are applied in order
to reduce the number of computationally expensive response analyses needed for
reliability estimation.
  We solve this problem by reformulating it as a linear programming problem.
Rigorous convergence analysis is performed, both in terms of convergence of
mean widths and in the sense of the Hausdorff metric. Additionally, numerical
examples are provided to illustrate the presented methods.
