This is the first of a two part paper investigating the geometry of the integrator reach sets, and the applications thereof. In this Part I, we establish that this compact convex set is semialgebraic, translated zonoid, and not a spectrahedron. We derive the parametric as well as the implicit representation of the boundary of this reach set. We also deduce the closed form formula for the volume and diameter of this set, and discuss their scaling with state dimension and time. We point out that these results may be utilized in benchmarking the performance of the reach set over-approximation algorithms.

中文翻译：

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集成商范围集的奇怪案例，第一部分：基本理论

这是研究集成器范围集的几何及其应用的两部分论文中的第一篇。在第一部分中，我们确定该紧凸集是半代数的，平移的带状体，而不是谱面体。我们导出此覆盖范围边界的参数以及隐式表示。我们还推导了该集合的体积和直径的封闭形式公式，并讨论了它们随状态维度和时间的缩放。我们指出，这些结果可用于基准测试范围集过近似算法的性能。