On generalized disc-polygons in plane convex bodies with a higher degree of smoothness
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866918141130440704 |
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| author | Fodor, Ferenc Papvári, Dániel I. |
| author_facet | Fodor, Ferenc Papvári, Dániel I. |
| contents | We prove power series expansions for the expectations of the number of vertices and missed area of random $L$-convex polygons in planar convex bodies with sufficiently smooth boundaries. Random $L$-convex polygons arise as the intersection of all translates of a fixed convex set $L$ that contain i.i.d. uniform random points from a suitable plane convex body $K$. Our results extend the asymptotic formulas proved in Fodor, Papvári and Vígh (2020) and Fodor and Montenegro (2024), and have consequences about $L$-convex floating bodies and relative affine surface area that were investigated by Schütt, Werner and Yalikun (2025). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_11702 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On generalized disc-polygons in plane convex bodies with a higher degree of smoothness Fodor, Ferenc Papvári, Dániel I. Metric Geometry We prove power series expansions for the expectations of the number of vertices and missed area of random $L$-convex polygons in planar convex bodies with sufficiently smooth boundaries. Random $L$-convex polygons arise as the intersection of all translates of a fixed convex set $L$ that contain i.i.d. uniform random points from a suitable plane convex body $K$. Our results extend the asymptotic formulas proved in Fodor, Papvári and Vígh (2020) and Fodor and Montenegro (2024), and have consequences about $L$-convex floating bodies and relative affine surface area that were investigated by Schütt, Werner and Yalikun (2025). |
| title | On generalized disc-polygons in plane convex bodies with a higher degree of smoothness |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2509.11702 |