Probabilidades sobre cuerpos convexos y cilindros

Abstract

H. GIGER and H. HADWIGER [1968] and R.E. MILES have recently considered different questions related to lattices of figures (convex bodies, r-flats or convex cylinders) in En . These lattices are assumed generated by N independent figures which intersect a fixed sphere S of radius R as R and N tends to -- in such a way that N/R tends to a positive constant, called the density of the lattice. In this paper we prove: a) The result does not change if instead of the sphere S we consider a convex body of arbitrary shape, which expands to the whole space En; this is a consequence of our Lemma 2. b) This result is applied to Theorem 1 (distribution function of the number of cylinders of a lattice which are intersected by a convex body K0 placed at random in space), which is essentially due to MILES with different proof. c) Theorem 2 refers to lattices of convex cylinders ir. E3 crossed by an arbitrary convex cylinder and we find the Distribution function of the number of intersected cylinders