Search

We study the properties of sets $\Sigma$ which are the solutions of the maximal distance minimizer problem, i.e. of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^n$ satisfying the inequality \[ max_{y \in M} dist(y,\Sigma) \leq r \] for a given compact set $M \subset \mathbb{R}^n$ and some given $r > 0$. Such sets can be considered as the shortest networks of radiating Wi-Fi cables arriving to each customer (for the set $M$ of customers) at a distance at most $r$. In this paper we prove that any maximal distance minimizer $\Sigma \subset \mathbb{R}^n$ has at most $3$ tangent rays at each point and the angle between any two tangent rays at the same point is at least $2\pi/3$. Moreover, at the plane (for $n=2$) we show that the number of points with three tangent rays is finite and maximal distance minimizer is a finite union of simple curves with continuous one-sided tangents. All the results are proved for the more general class of local minimizers, i.e. sets which are optimal under a perturbation of a neighbourhood of their arbitrary point., Comment: This work is the advanced version of the work arXiv:1910.07630,2019

Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the length (one-dimensional Hausdorff measure $\H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, \Sigma), \] where $dist$ stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems.

Consider a compact $M \subset \mathbb{R}^d$ and $r > 0$. A maximal distance minimizer problem is to find a connected compact set $\Sigma$ of the minimal length, such that \[ \max_{y \in M} dist (y, \Sigma) \leq r. \] The inverse problem is to determine whether a given compact connected set $\Sigma$ is a minimizer for some compact $M$ and some positive $r$. Let a Steiner tree $St$ with $n$ terminals be unique for its terminal vertices. The first result of the paper is that $St$ is a minimizer for a set $M$ of $n$ points and a small enough positive $r$. It is known that in the planar case a general Steiner tree (on a finite number of terminals) is unique. It is worth noting that a Steiner tree on $n$ terminal vertices can be not a minimizer for any $n$ point set $M$ starting with $n = 4$; the simplest such example is a Steiner tree for the vertices of a square. Recall that a planar maximal distance minimizer is a finite union of simple curves. The second result is an example of a minimizer with an infinite number of corner points (points with two tangent rays which do not belong to the same line), which means that this minimizer can not be represented as a finite union of smooth curves. We also show that every injective $C^{1,1}$-curve $\Sigma$ is a minimizer for a small enough $r>0$ and $M = \overline{B_r(\Sigma)}$ (this follows from the proof of analogues result by Tilli on average distance minimizers).

We study the properties of sets $\Sigma$ which are the solutions of the maximal distance minimizer problem, id est of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $\Sigma \subset \mathbb{R}^2$ satisfying the inequality $\max_{y \in M} dist(y,\Sigma) \leq r$ for a given compact set $M \subset \mathbb{R}^2$ and some given $r > 0$. Such sets can be considered as the shortest networks of radiating cables arriving to each customer (from the set $M$ of customers) at a distance at most $r$. In this work it is proved that each maximal distance minimizer is a union of finite number of simple curves, having one-sided tangents at each point. Moreover the angle between these rays at each point of a maximal distance minimizer is greater or equal to $2\pi/3$. It shows that a maximal distance minimizer is isotopic to a finite Steiner tree even for a "bad" compact $M$, which differs it from a solution of the Steiner problem (there exists an example of a Steiner tree with an infinite number of branching points). Also we classify the behavior of a minimizer in a neighbourhood of an arbitrary point of $\Sigma$. In fact, all the results are proved for more general class of local minimizer, id est sets which are optimal in a neighbourhood of its arbitrary point., Comment: v1 in Russian

We prove that the set of $n$-point configurations for which the solution of the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations on which at least two locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially require rely upon the theory of subanalytic sets developed in~\cite{bierstone1988semianalytic}. Motivated by this approach we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replace by an arbitrary analytic Riemannian manifold $M$. In this setup we argue that the set of configurations possessing two locally-minimal trees of the same length either has the dimension $n\dim M-1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to abovementioned results, we study the set of set of $n$-point configurations for which there is a unique solution of the Steiner problem in $\mathbb{R}^d$. We show that this set is path-connected.

A curve $\theta$: $I\to E$ in a metric space $E$ equipped with the distance $d$, where $I\subset \R$ is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time $\{t_i\}_{i=1}^3\subset I$ with $t_1\leq t_2\leq t_3$ one has $d(\theta(t_3),\theta(t_2))\leq d(\theta(t_3),\theta(t_1))$. We prove that if $E$ is a finite-dimensional normed space with an arbitrary norm, the trace of $\theta$ is bounded, then $\theta$ has finite length, i.e. is rectifiable, thus answering positively the question raised in~\cite{Lemenant16sc-rectif}., Comment: 27 pages

We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are \textit{tangential} polygons) and compute their Morse indices. This setup is motivated by a number of results about critical points and Morse indices of the oriented area function defined on the configuration space of polygons with prescribed edge lengths (flexible polygons). As a by-product, we present an independent computation of the Morse index of the area function (obtained earlier by G. Panina and A. Zhukova).

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 figures