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 tighter and more general error bounds than the state of the art. 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：本文研究具有光滑边界的紧集$X\subset\mathbb{R}^m$在光滑映射$f:\mathbb{R}^m\to\mathbb{R}^n$下的像$f(X)\subset\mathbb{R}^n$的凸包估计问题。假设$f$为浸没映射，我们推导了$f(X)$的凸包与$X$边界上$M$个采样点$x_i$的像$f(x_i)$的凸包之间豪斯多夫距离的新界。将该结果应用于随机样本几何推断问题时，给出的误差界比现有最优结果更紧且更具普适性。本文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的应用。