Elliptic polytopes are convex hulls of several concentric plane ellipses in \({{\mathbb {R}}}^d\). They arise in applications as natural generalizations of usual polytopes. In particular, they define invariant convex bodies of linear operators, optimal Lyapunov norms for linear dynamical systems, etc. To construct elliptic polytopes one needs to decide whether a given ellipse is contained in the convex hull of other ellipses. We analyse the computational complexity of this problem and show that for \(d=2, 3\), it admits an explicit solution. For larger d, two geometric methods for approximate solution are presented. Both use the convex optimization tools. The efficiency of the methods is demonstrated in two applications: the construction of extremal norms of linear operators and the computation of the joint spectral radius/Lyapunov exponent of a family of matrices.

椭圆多面体是\({{\mathbb {R}}}^d\)中几个同心平面椭圆的凸包 。它们在应用中作为常见多胞形的自然概括而出现。特别是，它们定义了线性算子的不变凸体、线性动力系统的最优李亚普诺夫范数等。为了构造椭圆多面体，需要确定给定的椭圆是否包含在其他椭圆的凸包中。我们分析了这个问题的计算复杂度，并表明对于\(d=2, 3\)，它有一个显式解。对于较大的 d，提出了两种近似求解的几何方法。两者都使用凸优化工具。该方法的效率在两个应用中得到了证明：线性算子的极值范数的构造和矩阵族的联合谱半径/Lyapunov指数的计算。