Mixed Volumes and Slices of the Cube

We give a combinatorial interpretation for the mixed volumes of two adjacent slices from the unit cube in terms of a refinement of the Eulerian numbers. 1998 Academic Press Let C d denote the d-dimensional unit cube, that is, C d =[x # R d : 0x i 1]. All the volumes and mixed volumes which appear in this note will be normalized so that the volume of C d is given by V(C d)=d!. We consider slices from the unit cube C d where the cutting hyperplanes are orthogonal to the vector (1, ..., 1). The kth slice, denoted by T d k , is the set T d k =