Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915277497696256 |
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| author | Rataj, Jan Winter, Steffen Zähle, Martina |
| author_facet | Rataj, Jan Winter, Steffen Zähle, Martina |
| contents | We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_04801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals Rataj, Jan Winter, Steffen Zähle, Martina Probability Primary 28A80, Secondary 28A75, 53C65, 60D05, 60G57 We consider a class of random self-similar fractals based on code trees which includes random recursive, homogeneous and V-variable fractals and many more. For such random fractals we consider mean values of the Lipschitz-Killing curvatures of their parallel sets for small parallel radii. Under the uniform strong open set condition and some further geometric assumptions we show that rescaled limits of these mean values exist as the parallel radius tends to 0. Moreover, integral representations are derived for these limits which recover and extend those known in the deterministic case and certain random cases. Results on the mean Minkowski content are included as a special case and shown to hold under weaker geometric assumptions. |
| title | Mean Minkowski content and mean fractal curvatures of random self-similar code tree fractals |
| topic | Probability Primary 28A80, Secondary 28A75, 53C65, 60D05, 60G57 |
| url | https://arxiv.org/abs/2505.04801 |