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Area properties associated with a convex plane curve

  • Dong-Soo Kim , Young Ho Kim EMAIL logo , Hyeong-Kwan Ju and Kyu-Chul Shim
Published/Copyright: July 21, 2016
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Georgian Mathematical Journal
From the journal Georgian Mathematical Journal Volume 24 Issue 3

Abstract

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 △A⁢B⁢P. 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 ℝ2. 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.

Keywords: Triangle; area; parabola; strictly locally convex curve; plane curvature
MSC 2010: 53A04

Funding source: National Research Foundation of Korea

Award Identifier / Grant number: 2015020387

Award Identifier / Grant number: 2012R1A1A2042298

Funding statement: The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2015020387). The second author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012R1A1A2042298).

Acknowledgements

The authors appreciate the anonymous referee for giving helpful suggestions to improve this paper.

References

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Received: 2014-6-6
Revised: 2014-12-30
Accepted: 2015-1-5
Published Online: 2016-7-21
Published in Print: 2017-9-1

© 2017 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Impulsive differential inclusions via variational method
  3. Frame properties of a part of an exponential system with degenerate coefficients in Hardy classes
  4. A note on two-variable Chebyshev polynomials
  5. Spectral analysis of dissipative fractional Sturm–Liouville operators
  6. Common best proximity pairs in strictly convex Banach spaces
  7. Traces of Muckenhoupt weighted function spaces in case of distant singularities
  8. The commutativity of prime Γ-rings with generalized skew derivations
  9. Δμ-sets and ∇μ-sets in generalized topological spaces
  10. On strong well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations with two independent variables
  11. Area properties associated with a convex plane curve
  12. Almost semi-correspondence
  13. The generalization of the Bernstein operator on any finite interval
  14. Some non-unique fixed point theorems of Ćirić type using rational-type contractive conditions
  15. On some properties of summability methods with variable order
  16. On the absolute convergence of Fourier series with respect to general orthonormal systems
https://doi.org/10.1515/gmj-2016-0027
Keywords for this article
Triangle; area; parabola; strictly locally convex curve; plane curvature

Articles in the same Issue

  1. Frontmatter
  2. Impulsive differential inclusions via variational method
  3. Frame properties of a part of an exponential system with degenerate coefficients in Hardy classes
  4. A note on two-variable Chebyshev polynomials
  5. Spectral analysis of dissipative fractional Sturm–Liouville operators
  6. Common best proximity pairs in strictly convex Banach spaces
  7. Traces of Muckenhoupt weighted function spaces in case of distant singularities
  8. The commutativity of prime Γ-rings with generalized skew derivations
  9. Δμ-sets and ∇μ-sets in generalized topological spaces
  10. On strong well-posedness of initial-boundary value problems for higher order nonlinear hyperbolic equations with two independent variables
  11. Area properties associated with a convex plane curve
  12. Almost semi-correspondence
  13. The generalization of the Bernstein operator on any finite interval
  14. Some non-unique fixed point theorems of Ćirić type using rational-type contractive conditions
  15. On some properties of summability methods with variable order
  16. On the absolute convergence of Fourier series with respect to general orthonormal systems
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