  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.
