On $L_p$-affine surface areas

Abstract

Let K be a convex body in R n with centroid at 0 and B be the Euclidean unit ball in R n centered at 0. We show that limt→0|K|-|K t | |B|-|B t | = O p (K) O p (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 {K t } t≥0 , {B t } 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].