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

TL;DR

This paper provides new error bounds for estimating the convex hull of a smooth image set, with applications in geometric inference, optimization, and dynamical systems, improving upon previous methods.

Contribution

It introduces a novel Hausdorff distance bound for convex hull approximation of smooth image sets, enhancing accuracy and generality in geometric inference.

Findings

Tighter error bounds for convex hull estimation from boundary samples.

Applicability to robust optimization and reachability analysis.

Improved bounds over previous methods in geometric inference.

Abstract

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.