  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 recently released 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 $10^{-27}$
- $10^{-28}$. 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 choice of such basis the flavor
ratio measured by IceCube can significantly change.
