On Lp-affine surface areas

Let K be a convex body in Rn with centroid at 0 and B be the Euclidean unit ball in Rn centered at 0. We show that limt→0 |K| − |Kt| |B| − |Bt| = Op(K) Op(B) , where |K| respectively |B| denotes the volume of K respectively B, Op(K) respectively Op(B) is the p-affine surface area of K respectively B and {Kt}t≥0, {Bt}t≥0 are general families of convex bodies constructed from K, B satisfying certain conditions. As a corollary we get results obtained in [23, 25, 26, 31].