Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 1
On a Dowker-Type Problem for Convex Disks with Almost Constant Curvature
Authors:
Bushra Basit Department of Algebra and Geometry, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary

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and
Zsolt Lángi Department of Algebra and Geometry, Budapest University of Technology and Economics, Műegyetem rkp. 3., H-1111 Budapest, Hungary
MTA-BME Morphodynamics Research Group, Műegyetem rkp. 3., H-1111 Budapest, Hungary

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Pages:
59–72
Online Publication Date:
15 Apr 2024
Publication Date:
15 Apr 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2024.04306
Keywords:
Dowker’s theorems; inscribed polygon; area; Hausdorff distance
Open access

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.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

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Studia Scientiarum Mathematicarum Hungarica
Language English
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Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
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Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
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ISSN 0081-6906 (Print)
ISSN 1588-2896 (Online)

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