On the Measure of line segments entirely contained in a convex body

Abstract

Let K be a convex body in the n-dimensional Euclidean space Rn. We consider the measure M0(l), in the sense of the Integral geometry (i.e. Invariant under the group of translations and rotations of Rn [6, Chap. 15]), of the set of non-oriented line segments of length l, which are entirely contained in K. This measure is related by (3.4) with the integrals Im for the power of the chords of K. These relations allow to obtain some inequalities, like (3.6), (3.7) and (3.8) for M0(l). Next we relate M0(l) with the function Ω(l) introduced by Enns and Ehlers [3], and prove a conjecture of these authors about the maximum of the average of the random straight line path through K. Finally, for n = 2, M0(l) is shown to be related by (5.6) with the associated function to K introduced by W. Pohl (S). Some representation formulas, like (3.9), (3.10) and (5.14) may be of independent interest