Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

SOME CHARACTERIZATIONS OF CONICS AND HYPERSURFACES WITH CENTRALLY SYMMETRIC HYPERPLANE SECTIONS

  • Shin-Ok Bang (Department of Mathematics Chonnam National University) ;
  • Dong Seo Kim (Department of Mathematics Chonnam National University) ;
  • Dong-Soo Kim (Department of Mathematics Chonnam National University) ;
  • Wonyong Kim (Department of Mathematics Chonnam National University)
  • Received : 2023.05.15
  • Accepted : 2023.06.16
  • Published : 2024.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

Parallel conics have interesting area and chord properties. In this paper, we study such properties of conics and conic hypersurfaces. First of all, we characterize conics in the plane with respect to the above mentioned properties. Finally, we establish some characterizations of hypersurfaces with centrally symmetric hyperplane sections.

Keywords

Acknowledgement

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2018R1D1A3B05050223).

References

  1. A. B'enyi, P. Szeptycki, and F. Van Vleck, Archimedean properties of parabolas, Amer. Math. Monthly 107 (2000), no. 10, 945-949. https://doi.org/10.2307/2695591 
  2. A. B'enyi, P. Szeptycki, and F. Van Vleck, A generalized Archimedean property, Real Anal. Exchange 29 (2003/04), no. 2, 881-889. 
  3. O. Ciaurri, E. Fernandez, and L. Roncal, Revisiting floating bodies, Expo. Math. 34 (2016), no. 4, 396-422. https://doi.org/10.1016/j.exmath.2016.06.001 
  4. D.-S. Kim, Ellipsoids and elliptic hyperboloids in the Euclidean space E n+1, Linear Algebra Appl. 471 (2015), 28-45. https://doi.org/10.1016/j.laa.2014.12.014 
  5. D.-S. Kim and S. H. Kang, A characterization of conic sections, Honam Math. J. 33 (2011), no. 3, 335-340. https://doi.org/10.5831/HMJ.2011.33.3.335 
  6. D.-S. Kim and Y. H. Kim, A characterization of ellipses, Amer. Math. Monthly 114 (2007), no. 1, 66-70. https://doi.org/10.1016/j.laa.2012.02.013 
  7. D.-S. Kim and Y. H. Kim, New characterizations of spheres, cylinders and W-curves, Linear Algebra Appl. 432 (2010), no. 11, 3002-3006. https://doi.org/10.1016/j.laa.2010.01.006 
  8. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120. https://doi.org/10.1016/j.laa.2012.02.013 
  9. D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl. 438 (2013), no. 3, 1356-1364. https://doi.org/10.1016/j.laa.2012.08.024 
  10. D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc. 50 (2013), no. 6, 2103-2114. https://doi.org/10.4134/BKMS.2013.50.6.2103 
  11. D.-S. Kim and Y. H. Kim, A characterization of concentric hyperspheres in Rn, Bull. Korean Math. Soc. 51 (2014), no. 2, 531-538. https://doi.org/10.4134/BKMS.2014.51.2.531 
  12. D.-S. Kim, J. H. Park, and Y. H. Kim Some characterizations of parabolas, Kyungpook Math. J. 53 (2013), no. 1, 99-104. https://doi.org/10.5666/KMJ.2013.53.1.99 
  13. D.-S. Kim, S. Park, and Y. H. Kim, Center of gravity and a characterization of parabolas, Kyungpook Math. J. 55 (2015), no. 2, 473-484. https://doi.org/10.5666/KMJ.2015.55.2.473 
  14. D.-S. Kim and K.-C. Shim, Area of triangles associated with a curve, Bull. Korean Math. Soc. 51 (2014), no. 3, 901-909. https://doi.org/10.4134/BKMS.2014.51.3.901 
  15. D.-S. Kim and B. Song, A characterization of elliptic hyperboloids, Honam Math. J. 35 (2013), no. 1, 37-49. https://doi.org/10.5831/HMJ.2013.35.1.37 
  16. J. Krawczyk, On areas associated with a curve, Zesz. Nauk. Uniw. Opol. Mat. 29 (1995), 97-101. 
  17. B. Richmond and T. A. Richmond, How to recognize a parabola, Amer. Math. Monthly 116 (2009), no. 10, 910-922. https://doi.org/10.4169/000298909X477023