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Calculation of Geometric Probabilities Using Covariogram of Convex Bodies

  • Stochastic and Integral Geometry
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Abstract

In the paper, a formula to calculate the probability that a random segment L(ω, u) in Rn with a fixed direction u and length l lies entirely in the bounded convex body D ⊂ Rn (n ≥ 2) is obtained in terms of covariogram of the body D. For any dimension n ≥ 2, a relationship between the probability P(L(ω, u) ⊂ D) and the orientation-dependent chord length distribution is also obtained. Using this formula, we obtain the explicit form of the probability P(L(ω, u) ⊂ D) in the cases where D is an n-dimensional ball (n ≥ 2), or a regular triangle on the plane.

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References

  1. L. A. Santalo, Integral Geometry and Geometric Probability (Addision-Wesley, Reading, 2004).

    Book  MATH  Google Scholar 

  2. R. De-Lin, Topics in Integral Geometry (Utopia, Singapore, 1994).

    Book  MATH  Google Scholar 

  3. G. Matheron, Random Sets and Integral Geometry (Wiley, New York, 1975).

    MATH  Google Scholar 

  4. G. Bianchi and G. Averkov, “Confirmation of Matheron’s Conjecture on the covariogram of a planar convex body”, Journal of the EuropeanMathematical Society 11, 1187–1202, 2009.

    MathSciNet  MATH  Google Scholar 

  5. R. Schneider and W. Weil, Stochastic and Integral Geometry (Springer, Berlin, 2008).

    Book  MATH  Google Scholar 

  6. V. K. Ohanyan and N. G. Aharonyan, “Tomography of bounded convex domains”, SUTRA: International Journal ofMathematical Sciences, 2 (1), 1–12, 2009.

    MATH  Google Scholar 

  7. N. G. Aharonyan, V. K. Ohanyan, “Chord lenght distribution functions for polygons”, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences) 40 (4), 43–56, 2005.

    Google Scholar 

  8. A. Gasparyan and V. K. Ohanyan, “Recognition of triangles by covariogram”, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 48 (3), 110–122, 2013.

    MATH  Google Scholar 

  9. H. S. Harutyunyan and V. K. Ohanyan, “Orientation-dependent section distributions for convex bodies”, Journal of ContemporaryMathematical Analysis (Armenian Academy of sciences), 49 (3), 139–156, 2014.

    MATH  Google Scholar 

  10. A. Gasparyan and V. K. Ohanyan, “Covariogramof a parallelogram”, Journal of ContemporaryMathematical Analysis (Armenian Academy of sciences), 49 (4), 194–206, 2014.

    MATH  Google Scholar 

  11. N. G. Aharonyan, H. S. Harutyunyan and V. K. Ohanyan, “Random copy of a segment within a convex domain”, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 45 (6), 348–356, 2010.

    MATH  Google Scholar 

  12. A. Gasparyan and V. K. Ohanyan, “Orientation-dependent distribution of the length of a random segment and covariogram”, Journal of Contemporary Mathematical Analysis (Armenian Academy of sciences), 50 (2), 90–97, 2015.

    MathSciNet  MATH  Google Scholar 

  13. N. G. Aharonyan and V. K. Ohanyan,“Kinematic measure of intervals lying in domains”, Journal of ContemporaryMathematical Analysis (Armenian Academy of sciences), 46 (5), 280–288, 2011.

    MATH  Google Scholar 

  14. S. Li, “Coincise formulas for the area and volume of a hyperspherical cap”, Asian J. of Mathematics and Statistics, 4 (1), 66–70, 2011.

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Yerevan State University, Yerevan, Armenia

    N. G. Aharonyan & V. K. Ohanyan

Authors
  1. N. G. Aharonyan
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  2. V. K. Ohanyan
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Corresponding authors

Correspondence to N. G. Aharonyan or V. K. Ohanyan.

Additional information

Original Russian Text © N. G. Aharonyan, V. K. Ohanyan, 2018, published in Izvestiya Natsional’noi Akademii Nauk Armenii, Matematika, 2018, No. 2, pp. 3–14.

The research of the second author was supported by theMathematical Studies Center at Yerevan State University.

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Aharonyan, N.G., Ohanyan, V.K. Calculation of Geometric Probabilities Using Covariogram of Convex Bodies. J. Contemp. Mathemat. Anal. 53, 113–120 (2018). https://doi.org/10.3103/S1068362318020061

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  • DOI: https://doi.org/10.3103/S1068362318020061

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