23 research outputs found
Du-Hwang Characteristic Area: Catch-22
The paper is devoted to description of two interconnected mistakes generated
by the gap in the Du and Hwang approach to Gilbert-Pollack Steiner ratio
conjecture.Comment: 4 pages, 2 figures, 10 ref
Minimal Spanning Trees on Infinite Sets
Minimal spanning trees on infinite vertex sets are investigated. A criterion
for minimality of a spanning tree having a finite length is obtained, which
generalizes the corresponding classical result for finite sets. It is given an
analytic description of the set of all infinite metric spaces which a minimal
spanning tree exists for. A sufficient condition for a minimal spanning tree
existence is obtained in terms of distances achievability between partitions
elements of the metric space under consideration. Besides, a concept of locally
minimal spanning tree is introduced, several properties of such trees are
described, and relations of those trees with (globally) minimal spanning trees
are investigated.Comment: 13 page
Dual Linear Programming Problem and One-Dimensional Gromov Minimal Fillings of Finite Metric Spaces
The present paper is devoted to studying of minimal parametric fillings of
finite metric spaces (a version of optimal connection problem) by methods of
Linear Programming. The estimate on the multiplicity of multi-tours appearing
in the formula of weight of minimal fillings is improved, an alternative proof
of this formula is obtained, and also explicit formulas for finite spaces
consisting of and points are derived.Comment: 19 pages, 4 figure
Optimal Networks
This mini-course was given in the First Yaroslavl Summer School on Discrete
and Computational Geometry in August 2012, organized by International Delaunay
Laboratory "Discrete and Computational Geometry" of Demidov Yaroslavl State
University. The aim of this mini-course is to give an introduction in Optimal
Networks theory. Optimal networks appear as solutions of the following natural
problem: How to connect a finite set of points in a metric space in an optimal
way? We cover three most natural types of optimal connection: spanning trees
connection without additional road forks, shortest trees and locally shortest
trees, and minimal fillings
Steiner Ratio for Manifolds
The Steiner ratio characterizes the greatest possible deviation of the length
of a minimal spanning tree from the length of the minimal Steiner tree. In this
paper, estimates of the Steiner ratio on Riemannian manifolds are obtained. As
a corollary, the Steiner ratio for flat tori, flat Klein bottles, and
projective plane of constant positive curvature are computed. Steiner ratio -
Steiner problem - Gilbert--Pollack conjecture - surfaces of constant curvatureComment: 11 page
Hausdorff Realization of Linear Geodesics of Gromov-Hausdorff Space
We have constructed a realization of rectilinear geodesic (in the sense
of~\cite{Memoli2018}), lying in Gromov-Hausdorff space, as a shortest geodesic
w.r.t. the Hausdorff distance in an ambient metric space.Comment: 5 pages, 1 figur
Geometry of Compact Metric Space in Terms of Gromov-Hausdorff Distances to Regular Simplexes
In the present paper we investigate geometric characteristics of compact
metric spaces, which can be described in terms of Gromov-Hausdorff distances to
simplexes, i.e., to finite metric spaces such that all their nonzero distances
are equal to each other. It turns out that these Gromov-Hausdorff distances
depend on some geometrical characteristics of finite partitions of the compact
metric spaces; some of the characteristics can be considered as a natural
analogue of the lengths of edges of minimum spanning trees. As a consequence,
we constructed an unexpected example of a continuum family of pairwise
non-isometric finite metric spaces with the same distances to all simplexes.Comment: 19 pages, 2 figure
The Gromov--Hausdorff Distance between Simplexes and Two-Distance Spaces
In the present paper we calculate the Gromov-Hausdorff distance between an
arbitrary simplex (a metric space all whose non-zero distances are the same)
and a finite metric space whose non-zero distances take two distinct values
(so-called -distance spaces). As a corollary, a complete solution to
generalized Borsuk problem for the -distance spaces is obtained. In
addition, we derive formulas for the clique covering number and for the
chromatic number of an arbitrary graph in terms of the Gromov-Hausdorff
distance between a simplex and an appropriate -distance space constructed by
the graph .Comment: 10 page
The Gromov-Hausdorff Distances between Simplexes and Ultrametric Spaces
In the present paper we investigate the Gromov--Hausdorff distances between a
bounded metric space and so called simplex, i.e., a metric space all whose
non-zero distances are the same. In the case when the simplex's cardinality
does not exceed the cardinality of , a new formula for this distance is
obtained. The latter permits to derive an exact formula for the distance
between a simplex and an ultrametric space.Comment: 10 page
Extendability of Metric Segments in Gromov--Hausdorff Distance
In this paper geometry of Gromov-Hausdorff distance on the class of all
metric spaces considered up to an isometry is investigated. For this class
continuous curves and their lengths are defined, and it is shown that the
Gromov-Hausdorff distance is intrinsic. Besides, metric segments are
considered, i.e., the classes of points lying between two given ones, and an
extension problem of such segments beyond their end-points is considered.Comment: 19 page