  Zonoids are Hausdorff limits of zonotopes, while zonotopes are convex
polytopes defined as the Minkowski sums of finitely many segments. We present a
combinatorial framework that links the study of mixed volumes of zonoids (a
topic that has applications in algebraic combinatorics) with the study of the
absolute value of the Grassmannian, defined as the image of the Grassmannian
under the coordinate-wise absolute value map. We use polyhedral computations to
derive new families of inequalities for n zonoids in dimension d, when
(n,d)=(6,2) and (6,3). Unlike the classical geometric inequalities, originating
from the Brunn-Minkowski and Aleksandrov-Fenchel inequalities, the inequalities
we produce have the special feature of being Minkowski linear in each of the n
zonoids they involve.
