We consider a continuous optimization model of a one-dimensional connected transportation network under the assumption that the cost of transportation with the use of network is negligible in comparison with the cost of transportation without it. We investigate the connections between this problem and its important special case, the minimization of the average distance functional. For the average distance minimization problem we formulate a number of conditions for the partial geometric regularity of a solution in R~n with an arbitrary dimension n ≥ 2. The corresponding results are applied to solutions to the general optimization problem.
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