  We consider $C^{1+\epsilon}$ diffeomorphisms of the torus, denoted $f,$
homotopic to the identity and whose rotation sets have interior. We give some
uniform bounds on the displacement of points in the plane under iterates of a
lift of $f,$ relative to vectors in the boundary of the rotation set and we use
these estimates in order to prove that if such a diffeomorphism $f$ preserves
area, then the rotation vector of the area measure is an interior point of the
rotation set. This settles a strong version of a conjecture proposed by P.
Boyland. We also present some new results on the realization of extremal points
of the rotation set by compact $f$-invariant subsets of the torus.
