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Some inequalities for chord power integrals of parallelotopes

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Abstract

We prove some geometric inequalities for pth-order chord power integrals \({\mathcal I}_p(P_d),\,1 \le p \le d,\) of d-parallelotopes \(P_d\) with positive volume \(V_d(P_d)\). First, we derive upper and lower bounds of the ratio \({\mathcal I}_p(P_d)/V_d^2(P_d)\) which are attained by a d-cuboid \(C_d\) with the same volume resp. the same mean breadth as \(P_d\). Second, we apply the device of Schur-convexity to obtain bounds of \({\mathcal I}_p(C_d)/V_d^2(C_d)\) which are attained by a d-cube with the same volume resp. the same mean breadth as \(C_d\). Most of these inequalities are shown for a more general class of ovoid functionals containing, as by-product, a Pfiefer-type inequality for d-parallelotopes.

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References

  1. Carleman, T.: Über eine isoperimetrische Aufgabe und ihre physikalischen Anwendungen. Math. Z. 3, 1-7 (1919)

    Article  MathSciNet  MATH  Google Scholar 

  2. Davy, P.J.: Inequalities for moments of secant length. Z. Wahrscheinlichkeitstheorie verw. Geb. 68, 243-246 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Florian, A.: Extremum problems for convex discs and polyhedra. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 177-221. Elsevier, North-Holland (1993)

    Chapter  Google Scholar 

  4. Gille, W.: Particle and Particle Systems Characterization–Small-Angle Scattering (SAS) Applications. CRC Press, Boca Raton (2014)

    MATH  Google Scholar 

  5. Hansen, J., Reitzner, M.: Electromagnetic wave propagation and inequalities for moments of chord lengths. Adv. Appl. Probab. 36, 987-995 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Heinrich, L.: Central limit theorems for Poisson hyperplanes in expanding convex bodies. Rend. Circ. Mat. Palermo Serie II(Suppl. 81), 187-212 (2009)

    MathSciNet  Google Scholar 

  7. Heinrich, L.: Some new results on second-order chord power integrals of convex quadrangles. Rend. Circ. Mat. Palermo Serie II(Suppl. 84), 195-205 (2012)

    MathSciNet  Google Scholar 

  8. Heinrich, L., Spiess, M.: Central limit theorem for volume and surface content of stationary Poisson cylinder processes in expanding domains. Adv. Appl. Probab. 45, 312-331 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Heinrich, L.: Lower and upper bounds for chord power integrals of ellipsoids. Appl. Math. Sci. 8(165), 8257-8269 (2014)

    Article  Google Scholar 

  10. Last, G., Penrose, M.D., Schulte, M., Thäle, C.: Moments and central limit theorems for some multivariate Poisson functionals. Adv. Appl. Probab. 46, 348-364 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Marshall, A.W., Olkin, I.: Inequalities: Theorie of Majorization and Its Applications. Academic Press, New York (1979)

    MATH  Google Scholar 

  12. Mitrinović, D.S., Pečarić, J.E., Volenec, V.: Recent Advances in Geometric Inequalities. Kluwer Academic Publishers, Dordrecht (1989)

    Book  MATH  Google Scholar 

  13. Pfiefer, R.E.: Maximum and minimum sets for some geometric mean values. J. Theor. Probab. 3, 169-179 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ren, D.: Topics in Integral Geometry. World Scientific Publishing, Singapore (1994)

    MATH  Google Scholar 

  15. Santaló, L.A.: Integral Geometry and Geometric Probability. Addison-Wesley, Reading (1976)

    MATH  Google Scholar 

  16. Schneider, R., Wieacker, J.A.: Integral geometry. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. B, pp. 1345-1390. North-Holland, Amsterdam (1993)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  18. Schreiber, T., Thäle, C.: Second-order theory for iteration stable tessellations. Probab. Math. Stat. 32, 281-300 (2012)

    MathSciNet  MATH  Google Scholar 

  19. Steele, J.M.: The Cauchy-Schwarz Master Class–An Introduction to the Art of Mathematical Inequalities. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  20. Xiong, G., Song, X.: Inequalities for chord power integrals. J. Korean Math. Soc. 45, 587-596 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Zhang, X.M.: Schur-convex functions and isoperimetric inequalities. Proc. Am. Math. Soc. 126, 461-470 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The author would like to thank the referee for his careful reading of the original manuscript.

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  1. Lothar Heinrich

    Present address: Institute of Mathematics, 86135, Augsburg, Germany

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  1. University of Augsburg, Augsburg, Germany

    Lothar Heinrich

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  1. Lothar Heinrich
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Correspondence to Lothar Heinrich.

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Communicated by A. Constantin.

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Heinrich, L. Some inequalities for chord power integrals of parallelotopes. Monatsh Math 181, 821–838 (2016). https://doi.org/10.1007/s00605-016-0888-y

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