Notes on random chords in convex bodies

Abstract

Consider a compact bodyEembedded in a convex compact bodyG.A pointPis randomly chosen inEand three different rays to the boundary∂GofGare generated byP.Rayis in a uniformly random direction and has lengthR, rayis through a second random point chosen from withinGand has lengthW, and rayis to a random point in∂Gand has lengthY.The distribution ofYis obtained and is related to previous work. WhenEis specialized toGor to∂G, other known results are retrieved. The paper ends with a discussion of a conjecture relating the means ofR, WandY.