Geometriae Dedicata

, Volume 121, Issue 1, pp 9–18 | Cite as

Recurrent approach to Blaschke’s problem

  • E. Gečiauskas
Original Paper


We have obtained a recurrence formula \(I_{n+3} = \frac{4(n+3)}{\pi(n+4)}VI_{n+1}\) for integro-geometric moments in the case of a circle with the area V, where \(I_n = \int \nolimits_{K \cap G}\sigma^{n}{\rm d} G\), while in the case of a convex domain K with the perimeter S and area V the recurrence formula
$$ I_{n+3} = \frac{8(n+3)V^2}{(n+1)(n+4)\pi}\Big[\frac{{\rm d} I_{n+1}}{{\rm d} V} - \frac{I_{n+1}}{S} \frac{{\rm d} S}{{\rm d} V} \Big] $$
holds, when curvature of the contour K(s) > 0,   n = 0,1,2,...


Moments of integral geometry Recurrence formula Blaschke’s problem 

Mathematics Subject Classification (2000)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blaschke W.: Vorlesungen uber integralgeometrie, 1 Heft, 2 Aufl. Berlin (1936)Google Scholar
  2. 2.
    Crofton M.W. (1877) Geometrical theorems related to mean values. Proc. Lond. Math. Soc. 8, 304–309CrossRefGoogle Scholar
  3. 3.
    Blaschke W. (1948) Eine isoperimetrische Eigenschaft des Kreises. Math. Z. B. 1, 52–57Google Scholar
  4. 4.
    Carleman T. (1919) Uber eine isoperimetrische Aufgabe und ihre physikalishen Anwendungen. Math. Z. B. 3, 1–8CrossRefMathSciNetGoogle Scholar
  5. 5.
    Blaschke, W.: Vorlesungen uber Integralgeometrie. 3 Aufl. Berlin (1955)Google Scholar
  6. 6.
    Voss K. (1982) Powers of chords for convex sets. Biom. J. 24(5): 513–516MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Piefke F. (1979) The chord length distribution of the ellipse. Lietuvos Matem. Rink. 19(3): 45–53MATHMathSciNetGoogle Scholar
  8. 8.
    Sulanke R.(1961) Die Verteilung der Sechnenlangen an ebenen und raumlichen Figuren. Math. Nachr. 23(1): 51–74MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gečiauskas E. (1994) On the second moment in Blaschke’s problem. Lith. Math. J. 34, 122–125CrossRefGoogle Scholar
  10. 10.
    Gečiauskas E. (1997) Fragments related with Blaschke’s problem. Lith. Math. J. 37, 246–248CrossRefGoogle Scholar
  11. 11.
    Mathai A.M., Pederzoli G. (1997) Random points with reference to a circle revisited. Rendiconti del circolo matematico di Palermo, Serie II, Suppl. 50, 235–258MathSciNetGoogle Scholar
  12. 12.
    Gečiauskas E. (1968) The method of the integral geometry for finding the distribution functions of chord length of an oval and of distance in an oval. Lith. Math. J. 8, 237–241Google Scholar
  13. 13.
    Kendall, M.G., Moran, P.A.P.: Geometric Probability. Moscow (1972) (in Russian)Google Scholar
  14. 14.
    Santalo, L.A.: Introduction to Integral Geometry. Moscow (1956) (in Russian)Google Scholar
  15. 15.
    Enns E.G., Ehlers P.F., Stuhr S. (1981) Every body has its moments. Statist. Distrib. Sci. Work 5, 387–396MathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Institute of Mathematics and Informatics Akademijos 4VilniusLithuania

Personalised recommendations