Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 61: Issue 3
Ball Characterizations in Planes and Spaces of Constant Curvature, I
Authors:
Jesús Jerónimo-Castro Facultad de Ingeniería, Universidad Autónoma de Querétaro, Centro Universitario, Cerro de las Campanas s/n C.P. 76010, Santiago de Querétaro, Qro. México, MEXICO

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and
Endre Makai Jr. Alfréd Rényi Institute of Mathematics, Hungarian Research Network (HUN-REN), H-1364 Budapest, Pf. 127, Hungary

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Pages:
274–310
Online Publication Date:
21 Nov 2024
Publication Date:
21 Nov 2024
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2024.04320
Keywords:
Spherical; Euclidean and hyperbolic planes; characterizations of circle/paracycle/hypercycle/half-plane; convex bodies; proper closed convex sets with interior points; directly congruent copies; intersections; central symmetry; axial symmetry
Restricted access

High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic plane. Let us have in 𝑆2, ℝ2 or 𝐻2 a pair of convex bodies (for 𝑆2 different from 𝑆2), such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any congruent copies of them are axially symmetric, then our bodies are (incongruent) circles. Let us have in 𝑆2, ℝ2 or 𝐻2 proper closed convex subsets 𝐾, 𝐿 with interior points, such that the numbers of the connected components of the boundaries of 𝐾 and 𝐿 are finite. If the intersections of any congruent copies of 𝐾 and 𝐿 are centrally symmetric, then 𝐾 and 𝐿 are congruent circles, or, for ℝ2, parallel strips. For ℝ2 we exactly describe all pairs of such subsets 𝐾, 𝐿, whose any congruent copies have an intersection with axial symmetry (there are five cases).

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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
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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ISSN 0081-6906 (Print)
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