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Area properties associated with a convex plane curve. (English) Zbl 1376.53006

Summary: Archimedes knew 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 chord \(AB\) is four thirds of the area of the triangle \(\bigtriangleup ABP\). Recently, the first two authors have proved that this fact is the characteristic property of parabolas. In this paper, we study strictly locally convex curves in the plane \(\mathbb{R}^2\). As a result, generalizing the above mentioned characterization theorem for parabolas, we present two conditions, which are necessary and sufficient, for a strictly locally convex curve in the plane to be an open arc of a parabola.

MSC:

53A04 Curves in Euclidean and related spaces
97G30 Area and volume (educational aspects)
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References:

[1] Á. Bényi, P. Szeptycki and F. Van Vleck, Archimedean properties of parabolas, Amer. Math. Monthly 107 (2000), no. 10, 945-949.; Bényi, Á.; Szeptycki, P.; Van Vleck, F., Archimedean properties of parabolas, Amer. Math. Monthly, 107, 10, 945-949 (2000) · Zbl 0986.51039
[2] Á. Bényi, P. Szeptycki and F. Van Vleck, A generalized Archimedean property, Real Anal. Exchange 29 (2004), no. 2, 881-889.; Bényi, Á.; Szeptycki, P.; Van Vleck, F., A generalized Archimedean property, Real Anal. Exchange, 29, 2, 881-889 (2004) · Zbl 1082.26001
[3] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, 1976.; do Carmo, M. P., Differential Geometry of Curves and Surfaces (1976) · Zbl 0326.53001
[4] W. A. Day, Inequalities for areas associated with conics, Amer. Math. Monthly 98 (1991), no. 1, 36-39.; Day, W. A., Inequalities for areas associated with conics, Amer. Math. Monthly, 98, 1, 36-39 (1991) · Zbl 0746.52011
[5] D.-S. Kim and S. H. Kang, A characterization of conic sections, Honam Math. J. 33 (2011), no. 3, 335-340.; Kim, D.-S.; Kang, S. H., A characterization of conic sections, Honam Math. J., 33, 3, 335-340 (2011) · Zbl 1231.53006
[6] D.-S. Kim, D. S. Kim, H. S. Bae and H.-J. Kim, Area of triangles associated with a strictly locally convex curve, Honam Math. J. 37 (2015), no. 1, 41-52.; Kim, D.-S.; Kim, D. S.; Bae, H. S.; Kim, H.-J., Area of triangles associated with a strictly locally convex curve, Honam Math. J., 37, 1, 41-52 (2015) · Zbl 1322.53005
[7] 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.; Kim, D.-S.; Kim, D. S.; Kim, Y. H., On triangles associated with a curve, Bull. Korean Math. Soc., 52, 3, 925-933 (2015) · Zbl 1343.53006
[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.; Kim, D.-S.; Kim, W.; Kim, Y. H.; Park, D. H., Area of triangles associated with a curve II, Bull. Korean Math. Soc., 52, 1, 275-286 (2015) · Zbl 1310.51023
[9] D.-S. Kim and Y. H. Kim, A characterization of ellipses, Amer. Math. Monthly 114 (2007), no. 1, 66-70.; Kim, D.-S.; Kim, Y. H., A characterization of ellipses, Amer. Math. Monthly, 114, 1, 66-70 (2007) · Zbl 1138.53006
[10] D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl. 437 (2012), no. 1, 113-120.; Kim, D.-S.; Kim, Y. H., Some characterizations of spheres and elliptic paraboloids, Linear Algebra Appl., 437, 1, 113-120 (2012) · Zbl 1248.53006
[11] D.-S. Kim and Y. H. Kim, On the Archimedean characterization of parabolas, Bull. Korean Math. Soc. 50 (2013), no. 6, 2103-2114.; Kim, D.-S.; Kim, Y. H., On the Archimedean characterization of parabolas, Bull. Korean Math. Soc., 50, 6, 2103-2114 (2013) · Zbl 1281.53004
[12] D.-S. Kim and Y. H. Kim, Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl. 438 (2013), no. 3, 1356-1364.; Kim, D.-S.; Kim, Y. H., Some characterizations of spheres and elliptic paraboloids II, Linear Algebra Appl., 438, 3, 1356-1364 (2013) · Zbl 1257.53012
[13] D.-S. Kim and J. H. Park, Some characterizations of parabolas, Kyungpook Math. J. 53 (2013), no. 1, 99-104.; Kim, D.-S.; Park, J. H., Some characterizations of parabolas, Kyungpook Math. J., 53, 1, 99-104 (2013) · Zbl 1286.53008
[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.; Kim, D.-S.; Shim, K.-C., Area of triangles associated with a curve, Bull. Korean Math. Soc., 51, 3, 901-909 (2014) · Zbl 1294.53004
[15] J. Krawczyk, On areas associated with a curve (in Polish), Zesz. Nauk. Uniw. Opol. Mat. 29 (1995), 97-101.; Krawczyk, J., On areas associated with a curve, Zesz. Nauk. Uniw. Opol. Mat., 29, 97-101 (1995) · Zbl 0871.53004
[16] B. Richmond and T. Richmond, How to recognize a parabola, Amer. Math. Monthly 116 (2009), no. 10, 910-922.; Richmond, B.; Richmond, T., How to recognize a parabola, Amer. Math. Monthly, 116, 10, 910-922 (2009) · Zbl 1229.26006
[17] S. Stein, Archimedes. What Did He Do Besides Cry Eureka?, Mathematical Association of America, Washington, 1999.; Stein, S., Archimedes. What Did He Do Besides Cry Eureka? (1999) · Zbl 0932.01011
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