We study the problem of estimating the convex hull of the image $f(X)\subset\mathbb{R}^n$ of a compact set $X\subset\mathbb{R}^m$ with smooth boundary through a smooth function $f:\mathbb{R}^m\to\mathbb{R}^n$. Assuming that $f$ is a submersion, we derive a new bound on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ sampled inputs $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from a random sample, our results give error bounds that are tighter and more general than in previous work. We present applications to the problems of robust optimization, of reachability analysis of dynamical systems, and of robust trajectory optimization under bounded uncertainty.

翻译：本文研究通过光滑函数 $f:\mathbb{R}^m\to\mathbb{R}^n$ 估计紧致集合 $X\subset\mathbb{R}^m$（具有光滑边界）的象 $f(X)\subset\mathbb{R}^n$ 的凸包问题。假设 $f$ 为浸没映射，我们推导出 $f(X)$ 的凸包与 $X$ 边界上 $M$ 个采样输入 $x_i$ 的象 $f(x_i)$ 的凸包之间豪斯多夫距离的新误差界。将该结果应用于随机样本的几何推断问题时，所得误差界较以往工作更为紧致且更具普适性。我们进一步展示了该方法在鲁棒优化、动力系统可达性分析以及有界不确定性下的鲁棒轨迹优化等问题中的应用。