Average-distance problem for parameterized curves

We consider approximating a measure by a parameterized curve subject to length penalization. That is for a given finite positive compactly supported measure $\mu$, for $p \geq 1$ and $\lambda>0$ we consider the functional $$E(\gamma) = \int_{\mathbb{R}^d} d(x, \Gamma_\gamma)^p d\mu(x) + \lambda \,\textrm{Length}(\gamma)$$ where $\gamma:I \to \mathbb{R}^d$, $I$ is an interval in $\mathbb{R}$, $\Gamma_\gamma = \gamma(I)$, and $d(x, \Gamma_\gamma)$ is the distance of $x$ to $\Gamma_\gamma$. The problem is closely related to the average-distance problem, where the admissible class are the connected sets of finite Hausdorff measure $\mathcal H^1$, and to (regularized) principal curves studied in statistics. We obtain regularity of minimizers in the form of estimates on the total curvature of the minimizers. We prove that for measures $\mu$ supported in two dimensions the minimizing curve is injective if $p \geq 2$ or if $\mu$ has bounded density. This establishes that the minimization over parameterized curves is equivalent to minimizing over embedded curves and thus confirms that the problem has a geometric interpretation

On the horseshoe conjecture for maximal distance minimizers

We study the properties of sets $\Sigma$ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality \mbox{max}_{y \in M} \mbox{dist}(y,\Sigma) \leq r for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$. Such sets can be considered shortest possible pipelines arriving at a distance at most $r$ to every point of $M$ which in this case is considered as the set of customers of the pipeline. We prove the conjecture of Miranda, Paolini and Stepanov about the set of minimizers for $M$ a circumference of radius $R>0$ for the case when $r < R/4.98$. Moreover we show that when $M$ is a boundary of a smooth convex set with minimal radius of curvature $R$, then every minimizer $\Sigma$ has similar structure for $r < R/5$. Additionaly we prove a similar statement for local minimizers.Comment: 25 pages, 21 figure

Approximation of length minimization problems among compact connected sets

In this paper we provide an approximation \`a la Ambrosio-Tortorelli of some classical minimization problems involving the length of an unknown one-dimensional set, with an additional connectedness constraint, in dimension two. We introduce a term of new type relying on a weighted geodesic distance that forces the minimizers to be connected at the limit. We apply this approach to approximate the so-called Steiner Problem, but also the average distance problem, and finally a problem relying on the p-compliance energy. The proof of convergence of the approximating functional, which is stated in terms of Gamma-convergence relies on technical tools from geometric measure theory, as for instance a uniform lower bound for a sort of average directional Minkowski content of a family of compact connected sets

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

<p>Given a finite measure µ with compact support, and λ > 0, the averagedistance problem, in the penalized formulation, is to minimize</p> <p>(0.1) E λ µ (Σ) := Z Rd d(x, Σ)dµ(x) + λH1 (Σ),</p> <p>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.</p