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.

翻译：我们提出了一种数值方法，用于计算平均宽度意义下的最小凸紧集 $\mathcal{B}\subset\mathbb{R}^N$。该最小化问题受限于约束：对于所有单位向量 $u\in S^{N-1}$，给定某个 Lipschitz 函数 $C$，需满足 $\max_{b\in\mathcal{B}}\langle b , u\rangle\geq C(u)$。该问题源于凸失效集假设下的环境包络构建。环境包络描述了极端环境条件，常用于海洋结构物设计初期的可靠性分析，旨在减少可靠性估计所需的计算密集型响应分析次数。我们通过将该问题重新表述为线性规划问题进行求解，并开展了严格的收敛性分析，既涉及平均宽度的收敛性，也涵盖 Hausdorff 度量意义下的收敛性。此外，通过数值算例对所述方法进行了验证。