Random approximation of convex bodies in Hausdorff metric
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866908497590878208 |
|---|---|
| author | Prochno, Joscha Schütt, Carsten Sonnleitner, Mathias Werner, Elisabeth M. |
| author_facet | Prochno, Joscha Schütt, Carsten Sonnleitner, Mathias Werner, Elisabeth M. |
| contents | While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant $K_n$ is given by the convex hull of $n$ independent random points chosen uniformly on the boundary or in the interior of $K$. When $K$ is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as $n\to\infty$. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02870 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Random approximation of convex bodies in Hausdorff metric Prochno, Joscha Schütt, Carsten Sonnleitner, Mathias Werner, Elisabeth M. Metric Geometry Probability 52A22, 52A27, 60D05 While there is extensive literature on approximation, deterministic as well as random, of general convex bodies $K$ in the symmetric difference metric, or other metrics arising from intrinsic volumes, very little is known for corresponding random results in the Hausdorff distance when the approximant $K_n$ is given by the convex hull of $n$ independent random points chosen uniformly on the boundary or in the interior of $K$. When $K$ is a polygon and the points are chosen on its boundary, we determine the exact limiting behavior of the expected Hausdorff distance between a polygon as $n\to\infty$. From this we derive the behavior of the asymptotic constant for a regular polygon in the number of vertices. |
| title | Random approximation of convex bodies in Hausdorff metric |
| topic | Metric Geometry Probability 52A22, 52A27, 60D05 |
| url | https://arxiv.org/abs/2404.02870 |