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Geometriae Dedicata

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

Recurrent approach to Blaschke’s problem

  • E. Gečiauskas
Original Paper

Abstract

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,...

Keywords

Moments of integral geometry Recurrence formula Blaschke’s problem 

Mathematics Subject Classification (2000)

52 

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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

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

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