# Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications

arXiv: https://arxiv.org/abs/2302.13970 · HTML: https://mlsanity.com/paper/2302.13970/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2023.

## Built on by (most cited first, 1 of 1)

- [Convex Hulls of Reachable Sets](https://mlsanity.com/paper/2303.17674/) (2023)

## Abstract (arXiv)

We study the problem of estimating the convex hull of the image f(X)subsetℝn of a compact set Xsubsetℝm with smooth boundary through a smooth function f:ℝm→ℝn. 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(xi) of M sampled inputs xi 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.