Journal of Mathematical Sciences, Vol. 132, No. 4, 2006 PARTIAL GEOMETRIC REGULARITY OF SOME OPTIMAL CONNECTED TRANSPORTATION NETWORKS E. O. Stepanov e.stepanov@sns.it UDC 517.9 We consider a continuous optimization model of a one-dimensional connected trans- portation 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 investi- gate the connections between this problem and its important special case, the minimiza- tion 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. Bibliography: 26 titles. Illustrations: 1 Figure. § 1. Introduction We assume that the distribution of the population in some region (city) is determined by a nonnegative finite Borel measure ϕ + with compact support in R n . It is required to find an optimal transportation net- work (schemes of urban public transport and/or underground) which could be the most convenient for the population to reach service centers and working places provided that the distribution of working places and service centers is determined by a nonnegative finite Borel measure ϕ - with compact support in R n .A given function A: R + → R + is interpreted as the effective cost of the movement of every citizen without using the transportation network (i.e., “on foot” or by their own transport”), so that the cost for covering the distance t is A(t ). In this paper, we consider a simplified model. The network is simulated by a closed connected set Σ ⊂ R n and the cost for movement with the use of the transportation network is assumed to be zero. The corresponding optimization problem is formulated as follows. Problem 1.1. Find a set (an optimal transportation network) Σ = Σ opt ⊂ R n minimizing the cost for movement of the population, provided that free travel over the network Σ → MK(ϕ + , ϕ - , Σ) is allowed, among all closed connected sets Σ ⊂ R n satisfying the length constraint H 1 (Σ)  l (l > 0 is given). The cost for movement MK(ϕ + , ϕ - , ·) is strictly defined with the use of the Monge–Kantorovich opti- mal mass transport problem (cf. details in [1–4]). We suggest two equivalent formulas for computing the Translated from Problemy Matematicheskogo Analiza, No. 31, 2005, pp. 129–157. 1072-3374/06/1324-0522 c  2006 Springer Science+Business Media, Inc. 522