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We use the expression Integral geometry in the sense given it by Blaschke in “Vorlesungen über Integrtüçeomelrie”. In a previous paper (“Integral formulas in Croflon style on the sphere and some inequalities referring to spherical curve” (1942), Duke Mathematical Journal, vol. 9, pp. 707) we generalized to the sphere many formulas of plane integral geometry and at the same time applied these to the demonstration of certain inequalities referring to spherical curves. The present paper considers analogous qüestions for surfaces of constant negative curvature and consequently for hyperbolic geometry https://doi.org/10.1215/S0012-7094-43-01064-6

We consider a countable number of independent random uniform lines in the hyperbolic plane (in the sense of the theory of geometrical probability) which divide the plane into an infinite number of convex polygonal regions. The main purpose of the paper is to compute the mean number of sides, the mean perímeter, the mean area and the second order moments of these quantities of such polygonal regions. For the Euclidean plane the problema has been considered by several authors, mainly Miles, who has taken it as the starting point of a series of papers which are the IMSÍS of the so-called stochastic geometry https://doi.org/10.2307/3212643

In the present paper we consider some points of the Integral Geometry in a Riemannian n-dimensional space and in elliptic and hyperbolic spaces