For many statistical experiments, there exists a multitude of optimal designs. If we consider models with uncorrelated observations and adopt the approach of approximate experimental design, the set of all optimal designs typically forms a multivariate polytope. In this paper, we mathematically characterize the polytope of optimal designs. In particular, we show that its vertices correspond to the so-called minimal optimum designs. Consequently, we compute the vertices for several classical multifactor regression models of the first and the second degree. To this end, we use software tools based on rational arithmetic; therefore, the computed list is accurate and complete. The polytope of optimal experimental designs, and its vertices, can be applied in several ways. For instance, it can aid in constructing cost-efficient and efficient exact designs.

翻译：对于许多统计实验，存在多种最优设计。如果我们考虑具有不相关观测的模型并采用近似实验设计的方法，所有最优设计的集合通常形成一个多变量多面体。本文在数学上刻画了最优设计的多面体。特别地，我们证明其顶点对应于所谓的最小最优设计。据此，我们计算了多个一阶和二阶经典多因子回归模型的顶点。为此，我们使用了基于有理运算的软件工具；因此，计算结果准确且完整。最优实验设计的多面体及其顶点有多种应用方式。例如，它有助于构建成本效益高且高效的精确实体设计。