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

On uniqueness in Steiner problem

We prove that the set of $n$-point configurations for which solution of the planar Steiner problem is not unique has Hausdorff dimension is at most $2n-1$. Moreover, we show that the Hausdorff dimension of $n$-points configurations on which some locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially requires some analytic structure and some finiteness, so that we prove a similar result for a complete Riemannian analytic manifolds under some apriori assumption on the Steiner problem on them

On regularity of maximal distance minimizers in Euclidean Space

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, in the plane (for $n=2$) we show that the number of points with three tangent rays is finite and every maximal distance minimizer is a finite union of simple curves with one-sided tangents continuous from the corresponding side. 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:1910.07630,201

An overview of maximal distance minimizers problem

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

An example of an infinite Steiner tree connecting an uncountable set

We construct an example of a Steiner tree with an infinite number of branching points connecting an uncountable set of points. Such a tree is proven to be the unique solution to a Steiner problem for the given set of points. As a byproduct we get the whole family of explicitly defined finite Steiner trees, which are unique connected solutions of the Steiner problem for some given finite sets of points, and with growing complexity (i.e. the number of branching points)

On homogeneous statistical distributions exoplanets for their dynamic parameters

Correct distributions of extrasolar systems for their orbital parameters (semi-major axes, period, eccentricity) and physical characteristics (mass, spectral type of parent star) are received. Orbital resonances in extrasolar systems are considered. It is shown, that the account of more thin effects, including with use of wavelet methods, in obviously incorrectly reduced distributions it is not justified, to what the homogeneous statistical distributions for dynamic parameters of exoplanets, received in the present work, testify.Comment: 9 pages, 15 figures; International Conference "100 years since Tunguska phenomenon: Past, present and future", (June 26-28, 2008. Russia, Moscow), Lomonosov readings 2009 (Moscow State University

Dynamic Resonance Effects in the Statistical Distributions of Asteroids and Comets

Some principles in the distribution of Centaurs and the "Scattered Disk" objects, as well as the Kuiper belt objects for its semi-major axes, eccentricities and inclinations of the orbits have been investigated. It has been established, that more than a half from them move on the resonant orbits and that is what has been predicted earlier. The divergence of the maximum in the observable distribution of the objects of the Kuiper belt for the semi-major axes with an exact orbital resonance has been interpreted.Comment: 7 pages, 5 figures, 1 table. International Conference "100 years since Tunguska phenomenon: Past, present and future", (June 26-28, 2008. Russia, Moscow), International Conference "Modern problems of astronomy" (August 12-18, 2007, Ukraine, Odessa