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http://dx.doi.org/10.4134/BKMS.b210311

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)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.2, 2022 , pp. 407-417 More about this Journal
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
Triangle; area; parabola; strictly convex curve; plane curvature; quadric; Archimedean characterization of parabolas;
Citations & Related Records
Times Cited By KSCI : 5  (Citation Analysis)
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1 M. P. do Carmo, Differential Geometry of Curves and Surfaces, translated from the Portuguese, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1976.
2 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   DOI
3 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   DOI
4 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   DOI
5 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   DOI
6 D.-S. Kim, D. S. Kim, and Y. H. Kim, On triangles associated with a curve, Bull. Korean Math. Soc. 52 (2015), no. 3, 925-933. https://doi.org/10.4134/BKMS.2015.52.3.925   DOI
7 D.-S. Kim, I. Kim, and Y. H. Kim, Volume properties and some characterizations of ellipsoids in 𝔼n+1, Turkish J. Math. 45 (2021), no. 2, 896-908. https://doi.org/10.3906/mat-2009-36   DOI
8 D.-S. Kim, W. Kim, Y. H. Kim, and D. H. Park, Area of triangles associated with a curve II, Bull. Korean Math. Soc. 52 (2015), no. 1, 275-286. https://doi.org/10.4134/BKMS.2015.52.1.275   DOI
9 D.-S. Kim and I. Kim, Chord and area properties of strictly convex curves, Comm. Korean Math. Soc. 36 (2021), no. 4, 801-815. https://doi.org/10.4134/CKMS.c200314   DOI
10 S. Stein, Archimedes. What did he do besides cry Eureka?, Mathematical Association of America, Washington, DC, 1999.
11 D.-S. Kim, Y. H. Kim, and Y.-T. Jung, Area properties of strictly convex curves, Math. 7 (2019), 1-18. https://doi.org/10.3390/math7050391   DOI
12 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   DOI