An overview of maximal distance minimizers problem (original) (raw)
Related papers
Approximation of Length Minimization Problems Among Compact Connected Sets
Siam Journal on Mathematical Analysis, 2015
In this paper we provide an approximationà 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 Γ-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.
Inverse maximal distance minimizer problem in a finite setup
Cornell University - arXiv, 2022
Consider a compact M ⊂ R d and r > 0. A maximal distance minimizer problem is to find a connected compact set Σ of the minimal length, such that max y∈M dist (y, Σ) ≤ r. The inverse problem is to determine whether a given compact connected set Σ 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 Σ is a minimizer for a small enough r > 0 and M = Br(Σ) (this follows from the proof of analogues result by Tilli on average distance minimizers).
Fermat–Steiner problem in the metric space of compact sets endowed with Hausdorff distance
Journal of Geometry, 2016
Fermat-Steiner problem consists in finding all points in a metric space Y such that the sum of distances from each of them to the points from some fixed finite subset of Y is minimal. This problem is investigated for the metric space Y = H(X) of compact subsets of a metric space X, endowed with the Hausdorff distance. For the case of a proper metric space X a description of all compacts K ∈ H(X) which the minimum is attained at is obtained. In particular, the Steiner minimal trees for three-element boundaries are described. We also construct an example of a regular triangle in H(R 2), such that all its shortest trees have no "natural" symmetry.
On the horseshoe conjecture for maximal distance minimizers
arXiv (Cornell University), 2015
We study the properties of sets Σ having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets Σ ⊂ R 2 satisfying the inequality max y∈M dist (y, Σ) ≤ r for a given compact set M ⊂ R 2 and some given r > 0. Such sets play the role of shortest possible pipelines arriving at a distance at most r to every point of M , where M is the set of customers of the pipeline. We prove a conjecture of Miranda, Paolini and Stepanov that describes 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 the boundary of a smooth convex set with minimal radius of curvature R, then every minimizer Σ has similar structure for r < R/5. Additionaly, we prove a similar statement for local minimizers.
This study provides a clear-cut solution to a minimum distance problem, in particular, the problem of finding the minimum distance from a point to a line to another point on the same side of the line. The straightforward solution is a Pythagorean relation or formula which can be derived through geometrical construction and reasoning, and analytical approach using differentiation, particularly, the application of extreme-value theorem. Such formula is vital in solving minimum distance problems with greater ease, accuracy and speed. This will lessen the cost and waste of materials in practical engineering and business applications.
Smallest enclosing ball multidistance
Communications in Information and Systems, 2012
The smallest enclosing ball problem is analyzed in the class of proper metric spaces. By using the diameter of the smallest enclosing ball of a set of points, we find conditions in order to ensure that the mentioned measure is a multidistance.
Qualitative Properties of Maximum Distance Minimizers and Average Distance Minimizers in \mathbb R n
Journal of Mathematical Sciences, 2000
We consider one-dimensional networks of finite length in R n minimizing the average distance functional and the maximum distance functional subject to the length constraint. Under natural conditions, such minimizers use maximum available length, cannot contain closed loops (i.e., homeomorphic images of a circumference S 1), and have some mild regularity properties. Bibliography: 11 titles.
On the general distance problem in RkR^kRk
On the general distance problem in RkR^kRk, 2022
Using the method of compression we obtain a generalized lower bound for the number of ddd-unit distances that can be formed from a set of nnn points in the euclidean space mathbbRk\mathbb{R}^kmathbbRk. By letting mathcalDn,d\mathcal{D}_{n,d}mathcalDn,d denotes the number of ddd-unit distances that can be formed from a set of nnn points in mathbbRk\mathbb{R}^kmathbbRk, then we obtain the lower bound \begin{align} \mathcal{D}_{n,d}\gg \frac{n\sqrt{k}}{d}.\nonumber \end{align}