On a Dowker-type problem for convex disks with almost constant curvature
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866910320403939328 |
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| author | Basit, Bushra Lángi, Zsolt |
| author_facet | Basit, Bushra Lángi, Zsolt |
| contents | A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex $n$-gons by disk-$n$-gons, obtained as the intersection of $n$ closed Euclidean unit disks. It has been proved recently that if $C$ is the unit disk of a normed plane, then the same properties hold for the area of $C$-$n$-gons circumscribed about a $C$-convex disk $K$ and for the perimeters of $C$-$n$-gons inscribed or circumscribed about a $C$-convex disk $K$, but for a typical origin-symmetric convex disk $C$ with respect to Hausdorff distance, there is a $C$-convex disk $K$ such that the sequence of the areas of the maximum area $C$-$n$-gons inscribed in $K$ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of $C$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_02378 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On a Dowker-type problem for convex disks with almost constant curvature Basit, Bushra Lángi, Zsolt Metric Geometry 52A40, 52A30, 52A27 A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body $K$, the areas of the maximum (resp. minimum) area convex $n$-gons inscribed (resp. circumscribed) in $K$ is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex $n$-gons by disk-$n$-gons, obtained as the intersection of $n$ closed Euclidean unit disks. It has been proved recently that if $C$ is the unit disk of a normed plane, then the same properties hold for the area of $C$-$n$-gons circumscribed about a $C$-convex disk $K$ and for the perimeters of $C$-$n$-gons inscribed or circumscribed about a $C$-convex disk $K$, but for a typical origin-symmetric convex disk $C$ with respect to Hausdorff distance, there is a $C$-convex disk $K$ such that the sequence of the areas of the maximum area $C$-$n$-gons inscribed in $K$ is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of $C$. |
| title | On a Dowker-type problem for convex disks with almost constant curvature |
| topic | Metric Geometry 52A40, 52A30, 52A27 |
| url | https://arxiv.org/abs/2308.02378 |