# Properties of Minimizers of Average-Distance Problem via Discrete Approximation of Measures

Given a finite measure µ with compact support, and λ > 0, the averagedistance problem, in the penalized formulation, is to minimize

(0.1) E λ µ (Σ) := Z Rd d(x, Σ)dµ(x) + λH1 (Σ),

among pathwise connected, closed sets, Σ. Here d(x, Σ) is the distance from a point to a set and H1 is the 1-Hausdorff measure. In a sense the problem is to find a onedimensional measure that best approximates µ. It is known that the minimizer Σ is topologically a tree whose branches are rectifiable curves. The branches may not be C 1 , even for measures µ with smooth density. Here we show a result on weak second-order regularity of branches. Namely we show that arc-length-parameterized branches have BV derivatives and provide a priori estimates on the BV norm. The technique we use is to approximate the measure µ, in the weak-∗ topology of measures, by discrete measures. Such approximation is also relevant for numerical computations. We prove the stability of the minimizers in appropriate spaces and also compare the topologies of the minimizers corresponding to the approximations with the minimizer corresponding to µ.