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We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebras, where we recover the zero-completions of Bandelt and Meletiou. Along the way, we prove various properties of halfspaces in finite rank median spaces and a duality result for locally convex median spaces., Comment: 39 pages, 3 figures. Compared to published version, corrected Figure 1 (needed in an example). Many thanks to Mark Hagen for spotting this

We introduce the so--called doubling metric on the collection of non--empty bounded open subsets of a metric space. Given a subset $U$ of a metric space $X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all open balls contained inside $U$, and taking their union. If $U$ is open, the predecessor of $U$ is an open set containing $U$. The directed doubling distance between $U$ and another subset $V$ is the number of times that the predecessor operation needs to be applied to $U$ to obtain a set that contains $V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the directed distance between $U$ and $V$ and the directed distance between $V$ and $U$., Comment: 22 pages

We deduce from Sageev's results that whenever a group acts locally elliptically on a finite dimensional CAT(0) cube complex, then it must fix a point. As an application, we give an example of a group G such that G does not have property (T), but G and all its finitely generated subgroups can not act without a fixed point on a finite dimensional CAT(0) cube complex, answering a question by Barnhill and Chatterji., 4 pages

Let $M$ be a differentiable manifold and $TM$ be its tangent bundle. A $C^0$-Finsler structure on $M$ is a continuous function $F: TM \rightarrow \mathbb R$ such that its restriction to each tangent space is a norm. In this work we present a large family of projectively equivalent $C^0$-Finsler manifolds $(\hat M \cong \mathbb R^2,\hat F)$. Their structures $\hat F$ don't have partial derivatives and they aren't invariant by any transformation group of $\hat M$. For every $p,q \in (\hat M,\hat F)$, we determine the unique minimizing path connecting $p$ and $q$. They are line segments parallel to the vectors $(\sqrt{3}/2,1/2)$, $(0,1)$ or $(-\sqrt{3}/2,1/2)$, or else a concatenation of two of these line segments. Moreover $(\hat M,\hat F)$ aren't Busemann $G$-spaces and they don't admit any bounded open $\hat F$-strongly convex subsets. Other geodesic properties of $(\hat M,\hat F)$ are also studied., 26 pages, accepted in Tohoku Mathematical Journal

We give a characterization for asymptotic dimension growth. We apply it to CAT(0) cube complexes of finite dimension, giving an alternative proof of N. Wright's result on their finite asymptotic dimension. We also apply our new characterization to geodesic coarse median spaces of finite rank and establish that they have subexponential asymptotic dimension growth. This strengthens a recent result of J. Spakula and N. Wright.

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted volume with minimum weighted perimeter. According to Chambers' recent proof of the log-convex density conjecture, for many densities on $\mathbb{R}^n$ the answer is a sphere about the origin. We seek to generalize his results to some other spaces of revolution or to two different densities for volume and perimeter. We provide general results on existence and boundedness and a new approach to proving circles about the origin isoperimetric., Comment: 17 pages, 1 figure

A metric space (X,d) is called Ptolemaic space (PT space) if the inequality |xy||zw|≤|xz||yw|+|xw||yz| holds for each quadruple of points x, y, z and w in X. Here |xy|:=d(x,y) denotes the distance of two points x and y. The main result proven in the paper is the following: If (X,d) is a PT space which is also a geodesic space (i.e., for each pair x,y there exists a geodesic connecting x and y) and if X is moreover homeomorphic to ℝ×[0,1] then X is isometric to a flat strip ℝ×[0,a]⊂ℝ 2 with its Euclidean metric. The authors also give a new short proof of the fact that a proper geodesic PT space is always strictly distance convex. (See also Th. Foertsch and V. Schroeder [Trans. Am. Math. Soc. 363, No. 6, 2891–2906 (2011; Zbl 1220.53092)]). This also indicates a positive answer to the open question if every proper geodesic PT space is also a CAT(0)-space.

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph X has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on X., 9 pp

Let X be a proper metric space. The sublinear Higson compactification hLX is a variant of the Higson compactification. Its boundary hLX\X is denoted νLX, and is called the sublinear Higson corona of X. The sublinear Higson corona is a functor from the category of coarse spaces to that of compact Hausdorff spaces. Let P be a compact metric space and X be an unbounded proper metric space. We show that the sublinear Higson corona of a product space P × X equipped with a cone metric is homeomorphic to a product P × νLX. Especially, the sublinear Higson corona of the n-dimensional Euclidean space is homeomorphic to the product of an (n − 1)-dimensional sphere and the sublinear Higson corona of natural numbers.

The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds., 31 pages, 16 figures. The new version is the extended journal submission; it includes additional material from a conference submission (ref. [6] in the paper)

We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm of constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space., 8 pages; references added, minor errors removed

We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median., 13 pages, 2 figures. To appear in Algebraic & Geometric Topology

The paper studies ways in which the sets of a partition of a lattice in $\RR^n$ become regular model sets. The main theorem gives equivalent conditions which assure that a matrix substitution system on a lattice in $\RR^n$ gives rise to regular model sets (based on $p$-adic-like internal spaces), and hence to pure point diffractive sets. The methods developed here are used to show that the $n-$dimensional chair tiling and the sphinx tiling are pure point diffractive., 29 pages, 7 figures

It is shown that there exists a sequence of 3-regular graphs $\{G_n\}_{n=1}^\infty$ and a Hadamard space $X$ such that $\{G_n\}_{n=1}^\infty$ forms an expander sequence with respect to $X$, yet random regular graphs are not expanders with respect to $X$. This answers a question of \cite{NS11}. $\{G_n\}_{n=1}^\infty$ are also shown to be expanders with respect to random regular graphs, yielding a deterministic sublinear time constant factor approximation algorithm for computing the average squared distance in subsets of a random graph. The proof uses the Euclidean cone over a random graph, an auxiliary continuous geometric object that allows for the implementation of martingale methods., incorporated Referees' comments

The {\em minimal unfolded region} (or the {\em heart}) of a bounded subset $\Om$ in the Euclidean space is a subset of the convex hull of $\Om$ the definition of which is based on reflections in hyperplanes. It was introduced to restrict the location of the points that give extreme values of certain functions, such as potentials whose kernels are monotone functions of the distance, and solutions of differential equations to which Aleksandrov's reflection principle can be applied. %the temperature of a heat conductor with some initial-boundary condition, in which case the points are called the hot spots. We show that the minimal unfolded regions of the convex hull and parallel bodies of $\Om$ are both included in that of $\Om$., Comment: to appear in Hokkaido Math. J

The basic object of the paper is the moduli space $M_{2,3}(L)$ of a closed polygonal linkage either in $\mathbb{R}^2$ or in $\mathbb{R}^3$. As was originally suggested by G. Khimshiashvili, the space $M_{2}(L)$ is equipped with the oriented area function $A$, whereas (as is suggested in the paper) $M_{3}(L)$ is equipped with the vector area function $S$. The latter are generically Morse functions, whose critical points have a nice description. In the preprint, we define a \textit{swap action} (that is, the action of some group generated by edge transpositions) on the space $M_{2,3}(L)$ which preserves the functions $A$ and $S$ and the Morse points. We prove that the commutant of the group acts trivially, present some computer experiments and formulate a conjecture.

This is the first of two papers which aim to understand quasi-isometries of a subclass of unimodular split solvable Lie groups. In the present paper, we show that locally (in a coarse sense), a quasi-isometry between two groups in this subclass is close to a map that respects their group structures., Comment: 48 pages (10pt, wide textwidth), 8 figures

The idea of CC-metric is introduced by Krause [4] and improved by Chen [1]. Later, CC-analogues of some of the topics that include the concept of CC-distance have been studied. In this work, we give the Chinese Checker version of Heron's Formula.

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic, then the $L_p$ compression of the wreath product $\mathbb{Z}\bwr H$ equals \[{\rm max}\left\{\frac{1}{p},\frac{1}{2}\right\}\] . We also show that the $L_p$ compression of $\mathbb{Z}\bwr \mathbb{Z}$ equals \[{\rm max}\left\{\frac{p}{2p-1},\frac23\right\} \] and that the $L_p$ compression of $(\mathbb{Z}\bwr\mathbb{Z})_0$ (the zero section of $\mathbb{Z}\bwr \mathbb{Z}$ , equipped with the metric induced from $\mathbb{Z}\bwr \mathbb{Z}$ ) equals \[{\rm max}\left\{\frac{p+1}{2p},\frac34\right\} \] . The fact that the Hilbert compression exponent of $\mathbb{Z}\bwr\mathbb{Z}$ equals $2/3$ while the Hilbert compression exponent of $(\mathbb{Z}\bwr\mathbb{Z})_0$ equals $3/4$ is used to show that there exists a Lipschitz function $f:(\mathbb{Z}\bwr\mathbb{Z})_0\to L_2$ which cannot be extended to a Lipschitz function defined on all of $\mathbb{Z}\bwr \mathbb{Z}$ .

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples.

A main tool in studying topological properties of sets definable in o-minimal structures is the Cell Decomposition Theorem. The present paper proposes its metric counterpart based on the idea of a Lipschitz cell. In contrast to earlier results, we give an algorithm of a Lipschitz cell decomposition involving only permutations of variables as changes of coordinates.

A simple plane closed curve $\Gamma$ satisfies the DNA Inequality if the average curvature of any closed curve contained inside $\Gamma$ exceeds the average curvature of $\Gamma$. In 1997 Lagarias and Richardson proved that all convex curves satisfy the DNA Inequality and asked whether this is true for any non-convex curve. They conjectured that the DNA Inequality holds for certain L-shaped curves. In this paper, we disprove this conjecture for all L-Shapes and construct a large class of non-convex curves for which the DNA Inequality holds. We also give a polynomial-time procedure for determining whether any specific curve in a much larger class satisfies the DNA Inequality., Comment: Versions 7--9 contains more figures, a summary of the proof, and other modifications. Version 6 has corrected a couple of minor notational problems with version 5. Versions 5--9 are (the same) major generalization of the theorem proved in versions 1--4

By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank $\geq 2$ equals 1. This answers a question of J-F. Lafont and S. Prassidis., Comment: 3 pages

We show that the interior of the convex core of a quasifuchsian punctured-torus group admits an ideal decomposition (usually an infinite triangulation) which is canonical in two different senses: in a combinatorial sense via the pleating invariants, and in a geometric sense via an Epstein-Penner convex hull construction in Minkowski space. The result extends to certain non-quasifuchsian punctured-torus groups, and in fact to all of them if a strong version of the Pleating Lamination Conjecture is true., 38 pages, 14 figures

Building on earlier work of Katětov, Uspenskij proved in [9] that the group of isometries of Urysohn’s universal metric space U, endowed with the pointwise convergence topology, is a universal Polish group (i.e it contains an isomorphic copy of any Polish group). Answering a question of Gao and Kechris, we prove here the following, more precise result: for any Polish group G, there exists a closed subset F of U such that G is topologically isomorphic to the group of isometries of U which map F onto itself.

J Hempel [Topology, 2001] showed that the set of distances of the Heegaard splittings (S,V, h^n(V)) is unbounded, as long as the stable and unstable laminations of h avoid the closure of V in PML(S). Here h is a pseudo-Anosov homeomorphism of a surface S while V is the set of isotopy classes of simple closed curves in S bounding essential disks in a fixed handlebody. With the same hypothesis we show the distance of the splitting (S,V, h^n(V)) grows linearly with n, answering a question of A Casson. In addition we prove the converse of Hempel's theorem. Our method is to study the action of h on the curve complex associated to S. We rely heavily on the result, due to H Masur and Y Minsky [Invent. Math. 1999], that the curve complex is Gromov hyperbolic., Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper2.abs.html

Given a matroid M on the ground set E, the Bergman fan B(M), or space of M-ultrametrics, is a polyhedral complex in R^E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics. Motivated by the phylogenetic situation, we study the following problem: Given a point w in R^E, we wish to find an M-ultrametric which is closest to it in the l_infty metric. The solution to this problem follows easily from the existence of the subdominant M-ultrametric: a componentwise maximum M-ultrametric which is componentwise smaller than w. A procedure for computing it is given, which brings together the points of view of matroid theory and tropical geometry. When the matroid in question is the graphical matroid of the complete graph K_n, the Bergman fan B(K_n) parameterizes the equidistant phylogenetic trees with n leaves. In this case, our results provide a conceptual explanation for Chepoi and Fichet's method for computing the tree that most closely matches measured data., Comment: 13 pages, 4 figures

For two convex bodies K and T in $R^n$, the covering number of K by T, denoted N(K,T), is defined as the minimal number of translates of T needed to cover K. Let us denote by $K^o$ the polar body of K and by D the euclidean unit ball in $R^n$. We prove that the two functions of t, N(K, tD) and N(D, tK^o), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K in $R^n$ and over positive integers n. In particular, this verifies the duality conjecture for entropy numbers of linear operators, posed by Pietsch in 1972, in the central case when either the domain or the range of the operator is a Hilbert space., Comment: 17 p., LATEX

We show that the symmetry groups of the cut cone Cut(n) and the metric cone Met(n) both consist of the isometries induced by the permutations on {1,...,n}; that is, Is(Cut(n))=Is(Met(n))=Sym(n) for n>4. For n=4 we have Is(Cut(4))=Is(Met(4))=Sym(3)xSym(4). This is then extended to cones containing the cuts as extreme rays and for which the triangle inequalities are facet-inducing. For instance, Is(Hyp(n))=Sym(n) for n>4, where Hyp(n) denotes the hypermetric cone., 8 pages, LaTeX, 2 postscript figures