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In this paper we provide an approximation \`a la Ambrosio-Tortorelli of someclassical minimization problems involving the length of an unknownone-dimensional set, with an additional connectedness constraint, in dimensiontwo. We introduce a term of new type relying on a weighted geodesic distancethat forces the minimizers to be connected at the limit. We apply this approachto approximate the so-called Steiner Problem, but also the average distanceproblem, and finally a problem relying on the p-compliance energy. The proof ofconvergence of the approximating functional, which is stated in terms ofGamma-convergence relies on technical tools from geometric measure theory, asfor instance a uniform lower bound for a sort of average directional Minkowskicontent of a family of compact connected sets.

Approximation of length minimization problems among compact connected sets