Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications
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| Format: | Preprint |
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2023
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| _version_ | 1866909131582996480 |
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| author | Lew, Thomas Bonalli, Riccardo Janson, Lucas Pavone, Marco |
| author_facet | Lew, Thomas Bonalli, Riccardo Janson, Lucas Pavone, Marco |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_13970 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications Lew, Thomas Bonalli, Riccardo Janson, Lucas Pavone, Marco Optimization and Control Computational Geometry Systems and Control Differential Geometry Probability Statistics Theory 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. |
| title | Estimating the Convex Hull of the Image of a Set with Smooth Boundary: Error Bounds and Applications |
| topic | Optimization and Control Computational Geometry Systems and Control Differential Geometry Probability Statistics Theory |
| url | https://arxiv.org/abs/2302.13970 |