  The distribution of the maximal relative height (MRH) of self-affine
one-dimensional elastic interfaces in a random potential is studied. We analyze
the ground state configuration at zero driving force, and the critical
configuration exactly at the depinning threshold, both for the random-manifold
and random-periodic universality classes. These configurations are sampled by
exact numerical methods, and their MRH distributions are compared with those
with the same roughness exponent and boundary conditions, but produced by
independent Fourier modes with normally distributed amplitudes. Using Pickands'
theorem we derive an exact analytical description for the right tail of the
latter. After properly rescaling the MRH distributions we find that corrections
from the Gaussian independent modes approximation are in general small, as
previously found for the average width distribution of depinning
configurations. In the large size limit all corrections are finite except for
the ground-state in the random-periodic class whose MRH distribution becomes,
for periodic boundary conditions, indistinguishable from the Airy distribution.
We find that the MRH distributions are, in general, sensitive to changes of
boundary conditions.
