Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
share
We study the problem of estimating the convex hull of the image f(X)⊂ℝ^n of a compact set X⊂ℝ^m with smooth boundary through a smooth function f:ℝ^m→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.
READ FULL TEXT