  We find that for small vacuum mixing angle $\theta$ and low energies ($s\ll
M^2_Z$) the width of matter, $d_{1/2}$, needed to have conversion probability
$P\geq 1/2$ should be larger than $d_{min}= \pi/(2\sqrt{2} G_{F} \tan 2
\theta)$: $d_{1/2}\geq d_{min}$. Here $G_F$ is the Fermi constant, $s$ is the
total energy squared in the center of mass and $M_Z$ is the mass of the $Z$
boson. The absolute minimum $d_{1/2}=d_{min}$ is realized for oscillations in a
uniform medium with resonance density. For all the other density distributions
(monotonically varying density, castle wall profile, etc.) the required width
$d_{1/2}$ is larger than $d_{min}$. The width $d_{min}$ depends on $s$, and for
$Z$-resonance channels at $s\sim M^2_Z$ we get that $d_{min}(s)$ is 20 times
smaller than the low energy value. We apply the minimum width condition, $d\geq
d_{min}$, to high energy neutrinos in matter as well as in neutrino background.
Using this condition, we conclude that the matter effect is negligible for
neutrinos propagating in AGN and GRBs environments. Significant conversion can
be expected for neutrinos crossing dark matter halos of clusters of galaxies
and for neutrinos produced by cosmologically distant sources and propagating in
the universe.
