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

Given a finite measure $\mu\geq 0$ with compact support, and $\lambda >0$, the average-distance problem, in the penalized formulation, is to minimize $E_\mu^\lambda(\Sigma):= \int_{\mathbb{R}^d} d(x,\Sigma)d\mu(x)+\lambda \mathcal{H}^1(\Sigma)$ among pathwise connected, closed sets, $\Sigma$. Here $d(x, \Sigma)$ is the distance from a point to a set and $\mathcal{H}^1$ is the 1-Hausdorff measure. In a sense the problem is to find a one-dimensional measure that best approximates $\mu$. The minimizer, $\Sigma$, is topologically a tree whose branches are rectifiable curves. The branches may not be $C^1$, even for measures $\mu$ 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 $\mu$, in the weak-$*$ topology of measures, by discrete measures. Such approximation is also relevant for numerical compu...