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