In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some
classical minimization problems involving the length of an unknown
one-dimensional set, with an additional connectedness constraint, in dimension
two. We introduce a term of new type relying on a weighted geodesic distance
that forces the minimizers to be connected at the limit. We apply this approach
to approximate the so-called Steiner Problem, but also the average distance
problem, and finally a problem relying on the p-compliance energy. The proof of
convergence of the approximating functional, which is stated in terms of
Gamma-convergence relies on technical tools from geometric measure theory, as
for instance a uniform lower bound for a sort of average directional Minkowski
content of a family of compact connected sets
