The distance between two random points in plane regions

Abstract

LetTbe a triangle.Pbe a parallelogram,Ebe an ellipse,A,Bbe concentric circles,C,Dbe concentric dartboard regions,R,Sbe rectangles of the same orientation,U, Vbe two finite unions and/or differences of convex regions in the Euclidean plane. Given a functionfon [0,∞), letE[/(r),U,V] denote the mean value off(|u–v|), where |u–v| is the distance betweenu∊Uandv∊V. Using Borel’s overlap technique, a specific distance weight function and a specific equivalence relation, we obtain formulae expressingE[f(r),U,V] in terms of triple integrals, expressingE(rn,U,V),E[f(r),A,V] andE[f(r),R,V] in terms of double integrals, expressingE[f(r),A,B],E[f(r),R,S], E[f(r),T,T],E[f(r),P,P],E(rn,C,D)andE(rn,R,V) in terms of single integrals, and expressingE(rn,R,S),E(rn,P,P),E(rn,T,T),E(rn,E,E) in terms of elementary functions, wherenis an integer ≧−1. Many other related results are also given.