  Among the information provided by high energy neutrinos, a promising
possibility is to analyze the effects of a Violation of Equivalence Principle
(VEP) on neutrino oscillations. We analyze the IceCube data on atmospheric
neutrino fluxes under the assumption of a VEP and obtain updated constraints on
the parameter space with the benchmark choice that neutrinos with different
masses couple with different strengths to the gravitational field. In this case
we find that the VEP parameters times the local gravitational potential at
Earth can be constrained at the level of $10^{-27}$. We show that the
constraints from atmospheric neutrinos strongly depend on the assumption that
the neutrino eigenstates interacting diagonally with the gravitational field
coincide with the mass eigenstates, which is not a priori justified: this is
particularly clear in the case that the basis of diagonal gravitational
interaction coincide with the flavor basis, which cannot be constrained by the
observation of atmospheric neutrinos. Finally, we quantitatively study the
effect of a VEP on the flavor composition of the astrophysical neutrinos,
stressing again the interplay with the basis in which the VEP is diagonal: we
find that for some choices of such basis the flavor ratio measured by IceCube
can significantly change.
