Gemischte volumina in Kanalscharen

A canal class of convex bodies inn-dimensional Euclidean space consists of all convex bodies which have the same orthogonal projection on some hyperplane. In such a canal class, improved versions of the general Brunn-Minkowski theorem and of the Aleksandrov-Fenchel inequalities for mixed volumes are valid. Partial results on the equality cases are obtained. As an application, a translation theorem of the Aleksandrov-Fenchel-Jessen type is proved.