Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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AREA PROPERTIES ASSOCIATED WITH STRICTLY CONVEX CURVES

  • Bang, Shin-Ok (Department of Mathematics Chonnam National University) ;
  • Kim, Dong-Soo (Department of Mathematics Chonnam National University) ;
  • Kim, Incheon (Department of Mathematics Chonnam National University)
  • Received : 2021.04.16
  • Accepted : 2021.07.23
  • Published : 2022.03.31
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Abstract

Archimedes proved that for a point P on a parabola X and a chord AB of X parallel to the tangent of X at P, the area of the region bounded by the parabola X and the chord AB is four thirds of the area of the triangle ∆ABP. This property was proved to be a characteristic of parabolas, so called the Archimedean characterization of parabolas. In this article, we study strictly convex curves in the plane ℝ2. As a result, first using a functional equation we establish a characterization theorem for quadrics. With the help of this characterization we give another proof of the Archimedean characterization of parabolas. Finally, we present two related conditions which are necessary and sufficient for a strictly convex curve in the plane to be an open arc of a parabola.

Keywords

Acknowledgement

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

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