Profile vectors in the lattice of subspaces

The profile vector f ( U ) ∈ R n + 1 of a family U of subspaces of an n -dimensional vector space V over G F ( q ) is a vector of which the i th coordinate is the number of subspaces of dimension i in the family U ( i = 0 , 1 , … , n ) . In this paper, we determine the profile polytope of intersecting families (the convex hull of the profile vectors of all intersecting families of subspaces).