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

Abstract

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 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 $\mu$ and obtain a topological lower-semicontinuity result.