Two applications of the integral geometry in affine and projective spaces

Abstract

The integral geometry in projective space was initiated by VARGA [1935] and continued, together with the integral geometry in affine space, by the present author [1950]. In this paper we give two applications of these concepts. First we consider the density for sets of pairs of parallel hyper planes invariant with respect to the unimodular affine group. Then we evaluate the measure of all pairs of parallel hyper planes which contain a given convex body K. The second application concerns the density for sets of hyper quadrics invariant with respect to the projective group. We give the explicit forms (5.14), (5.16) and (5. 19) of this density. For n = 2 (conics on the plane) the formula (5.19) was given by STOKA