A general geometric construction for affine surface area

Let K be a convex body in R and B be the Euclidean unit ball in R. We show that limt→0 |K| − |Kt| |B| − |Bt| = as(K) as(B) , where as(K) respectively as(B) is the affine surface area of K respectively B and {Kt}t≥0, {Bt}t≥0 are general families of convex bodies constructed from K, B satifying certain conditions. As a corollary we get results obtained in [M-W], [Schm],[S-W] and[W]. The affine surface area as(K) was introduced by Blaschke [B] for convex bodies in R with sufficiently smooth boundary and by Leichtweiss [L1] for convex bodies in R with sufficiently smooth boundary as follows