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 diffeomorphism or a submersion, we derive new bounds on the Hausdorff distance between the convex hull of $f(X)$ and the convex hull of the images $f(x_i)$ of $M$ samples $x_i$ on the boundary of $X$. When applied to the problem of geometric inference from random samples, 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：我们研究通过光滑函数 $f:\mathbb{R}^m\to\mathbb{R}^n$ 估计紧集 $X\subset\mathbb{R}^m$（具有光滑边界）的像 $f(X)\subset\mathbb{R}^n$ 的凸包问题。在假设 $f$ 为微分同胚或浸没的条件下，我们推导了 $f(X)$ 凸包与 $X$ 边界上 $M$ 个采样点 $x_i$ 的像 $f(x_i)$ 凸包之间的豪斯多夫距离的新界限。当将该结果应用于随机采样下的几何推断问题时，我们的误差界比现有最优结果更紧且更具一般性。我们展示了该成果在鲁棒优化、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问题中的应用。