We show that the fractal curvature measures of invariant sets of
one-dimensional conformal iterated function systems satisfying the open set
condition exist, if and only if the associated geometric potential function is
nonlattice. Moreover, in the nonlattice situation we obtain that the Minkowski
content exists and prove that the fractal curvature measures are constant
multiples of the δ-conformal measure, where δ denotes the
Minkowski dimension of the invariant set. For the first fractal curvature
measure, this constant factor coincides with the Minkowski content of the
invariant set. In the lattice situation we give sufficient conditions for the
Minkowski content of the invariant set to exist, contrasting the fact that the
Minkowski content of a self-similar lattice fractal never exists. However,
every self-similar set satisfying the open set condition exhibits a Minkowski
measurable C1+α diffeomorphic image. Both in the lattice
and nonlattice situation average versions of the fractal curvature measures are
shown to always exist.Comment: 36 page