Stephen Ducret, Marc Troyanov
The aim of this note is to prove that any compact metric space can be made connected at a minimal cost, where the cost is taken to be the one-dimensional Hausdorff measure.
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Among other results, we prove the following theorem about Steiner minimal trees in $d$-dimensional Euclidean space: if two finite sets in $\mathbb{R}^d$ have unique and combinatorially equivalent Steiner minimal trees, then there is a homotopy between the two sets that maintains the uniqueness and the combinatorial structure of the Steiner minimal tree throughout the homotopy.
Steiner Minimal Trees in the Gromov--Hausdorff Space and in Gromov--Hausdorff Clouds
该研究在格罗莫夫-豪斯多夫理论的两种情形下，分别证明了有限个度量空间族上施泰纳最小树的存在性，为相关几何问题提供了理论支撑。
In this note we prove two results on the quantitative illumination parameter f(d) of the unit ball of a d-dimensional normed space introduced by K. Bezdek (1992). The first is that f(d) = O(2^d d^2 log d). The second involves Steiner minimal trees. Let v(d) be the maximum degree of a vertex, and s(d) of a Steiner point, in a Steiner minimal tree in a d-dimensional normed space, where both maxima are over all norms. F. Morgan (1992) conjectured that s(d) <= 2^d, and D. Cieslik (1990) conjectured v(d) <= 2(2^d-1). We prove that s(d) <= v(d) <= f(d) which, combined with the above estimate of f(d)…
In the present paper we investigate the metric space $\cal M$ consisting of isometry classes of compact metric spaces, endowed with the Gromov-Hausdorff metric. We show that for any finite subset $M$ from a sufficiently small neighborhood of a generic finite metric space, providing $M$ consists of finite metric spaces with the same number of points, each Steiner minimal tree in $\cal M$ connecting $M$ is a minimal filling for $M$. As a consequence, we prove that the both Steiner ratio and Gromov-Steiner ratio of $\cal M$ are equal to $1/2$.
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 curvature
We introduce irreducible correspondences that enables us to calculate the Gromov--Hausdorff distances effectively. By means of these correspondences, we show that the set of all metric spaces each consisting of no more than $3$ points is isometric to a polyhedral cone in the space $R^3$ endowed with the maximum norm. We prove that for any $3$-point metric space such that all the triangle inequalities are strict in it, there exists a neighborhood such that the Steiner minimal trees (in Gromov-Hausdorff space) with boundaries from this neighborhood are minimal fillings, i.e., it is impossible to…
It is shown that each finite family of finite metric spaces, being considered as a subset of Gromov--Hausdorff space, can be connected by a Steiner minimal tree.
The 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 $K\in H(X)$ 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 $…
We find upper bounds for the degrees of vertices and Steiner points in Steiner Minimal Trees in the d-dimensional Banach spaces \ell_p^d independent of d. This is in contrast to Minimal Spanning Trees, where the maximum degree of vertices grows exponentially in d (Robins and Salowe, 1995). Our upper bounds follow from characterizations of singularities of SMT's due to Lawlor and Morgan (1994), which we extend, and certain \ell_p-inequalities. We derive a general upper bound of d+1 for the degree of vertices of an SMT in an arbitrary smooth d-dimensional Banach space; the same upper bound for S…
The present paper opens a new branch in the theory of variational problems with branching extremals, the investigation of one-dimensional minimal fillings of finite pseudo-metric spaces. On the one hand, this problem is a one-dimensional version of a generalization of Gromov's minimal fillings problem to the case of stratified manifolds (the filling in our case is a weighted graph). On the other hand, this problem is interesting in itself and also can be considered as a generalization of another classical problem, namely, the Steiner problem on the construction of a shortest network joining a …
This paper introduces an exact algorithm for the construction of a shortest curvature-constrained network interconnecting a given set of directed points in the plane and an iterative method for doing so in 3D space. Such a network will be referred to as a minimum Dubins network, since its edges are Dubins paths (or slight variants thereof). The problem of constructing a minimum Dubins network appears in the context of underground mining optimisation, where the aim is to construct a least-cost network of tunnels navigable by trucks with a minimum turning radius. The Dubins network problem is si…
We give a new proof that a star $\{op_i:i=1,...,k\}$ in a normed plane is a Steiner minimal tree of its vertices $\{o,p_1,...,p_k\}$ if and only if all angles formed by the edges at o are absorbing [Swanepoel, Networks \textbf{36} (2000), 104--113]. The proof is more conceptual and simpler than the original one. We also find a new sufficient condition for higher-dimensional normed spaces to share this characterization. In particular, a star $\{op_i: i=1,...,k\}$ in any CL-space is a Steiner minimal tree of its vertices $\{o,p_1,...,p_k\}$ if and only if all angles are absorbing, which in turn …
The Fermat--Steiner problem is to find all points of the metric space Y such that the sum of the distances from each of them to points from some fixed finite subset A = {A_1, ..., A_n} of the space Y is minimal. This problem is considered in the case when Y=H(X) is the space of non-empty compact subsets of a finite-dimensional normed space X endowed with the Hausdorff metric, i.e. H(X) is a hyperspace over X. The set A is called boundary, all A_i are called boundary sets, and the compact sets that realize the minimum of the sum of distances to A_i are called Steiner compacts. In this paper, we…
We develop a general method of proving that certain star configurations in finit e-dimensional normed spaces are Steiner minimal trees. This method generalizes the results of Lawlor and Morgan (1994) that could only be applied to differentiable norms. The generalization uses the subdifferential calculus from convex analysis. We apply this method to two special norms. The first norm, occurring in the work of Cieslik, has unit ball the polar of the difference body of the n-simplex (in dimension 3 this is the rhombic dodecahedron). We determine the maximum degree of a given point in a Steiner min…
We consider a general metric Steiner problem which is of finding a set $\mathcal{S}$ with minimal length such that $\mathcal{S} \cup A$ is connected, where $A$ is a given compact subset of a given complete metric space $X$; a solution is called Steiner tree. Paolini, Stepanov and Teplitskaya provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension which was an open question (the corresponding tree is a self-similar fractal).
On the Existence of Minimal Parametric Networks
研究度量空间中最小网络存在条件，引入紧装备伪度量空间概念，证明其条件足以保证最小参数网络存在，推广相关定理，还能更一般表述贝德诺夫定理，且展示了超空间继承该性质的情况。
In the present paper we show how one can calculate the lengths of edges of a minimum spanning tree constructed for a finite metric space, in terms of the Gromov-Hausdorff distances from this space to simplices of sufficiently large diameter. Here by simplices we mean finite metric spaces all of whose nonzero distances are the same. As an application, we reduce the problems of finding a Steiner minimal tree length or a minimal filling length to maximization of the total distance to some finite number of simplices considered as points of the Gromov-Hausdorff space.
The aim of this paper is to prove isoperimetric inequalities for simplices and polytopes with $d+2$ vertices in Euclidean, spherical and hyperbolic $d$-space. In particular, we find the minimal volume $d$-dimensional hyperbolic simplices and spherical tetrahedra of a given inradius. Furthermore, we investigate the properties of maximal volume spherical and hyperbolic polytopes with $d+2$ vertices with a given circumradius, and the hyperbolic polytopes with $d+2$ vertices with a given inradius and having a minimal volume or minimal total edge length. Finally, for any $1 \leq k \leq d$, we inves…
The course was given at Peking University, Fall 2019. We discuss the following subjects: (1) Introduction to general topology, hyperspaces, metric and pseudometric spaces, graph theory. (2) Graphs in metric spaces, minimum spanning tree, Steiner minimal tree, Gromov minimal filling. (3) Hausdorff distance, Vietoris topology, Limits theory, inheritance of completeness, total boundedness, compactness by hyperspaces. (4) Gromov-Hausdorff distance, triangle inequality, positive definiteness for isometry classes of compact spaces, counterexample for boundedly compact spaces. (5) Gromov-Hausdorff di…
Inverse maximal and average distance minimizer problems
本文研究了极大距离最小化问题的逆向问题，证明了唯一 Steiner 树、含无限角点的曲线及单射 $C^{1,1}$-曲线均可作为最小化子，并将 Tilli 的平均距离最小化子结果推广至 $d$ 维欧几里得空间。
The embedding of finite metrics in $\ell_1$ has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems in which there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into $\ell_1$. Here we show that this theory can be generalized considerably to encompass Steiner tree packing problems in both graphs and hypergraphs. Instead of the theory of $\ell_1$ metrics and minimal distortion embeddings, the parallel is the theory of diversities recently introduce…
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