Recurrent approach to Blaschke’s problem

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,...