Volume inequalities for the i-th-convolution bodies

We obtain a new extension of Rogers–Shephard inequality providing an upper bound for the volume of the sum of two convex bodies K and L. We also give lower bounds for the volume of the k-th limiting convolution body of two convex bodies K and L  . Special attention is paid to the (n−1)(n−1)-th limiting convolution body, for which a sharp inequality, which is equality only when K=−LK=−L is a simplex, is given. Since the n-th limiting convolution body of K and −K is the polar projection body of K, these inequalities can be viewed as an extension of Zhang's inequality.