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Register a new userSome weak indivisibility results in ultrahomogeneous metric spaces
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We study the validity of a partition property known as weak indivisibility for the integer and the rational Urysohn metric spaces. We also compare weak indivisiblity to another partition property, called age-indivisibility, and provide an example of a countable ultrahomogeneous metric space which may be age-indivisible but not weakly indivisible.
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An important problem that commonly arises in areas such as internet traffic-flow analysis, phylogenetics and electrical circuit design, is to find a representation of any given metric $D$ on a finite set by an edge-weighted graph, such that the total edge length of the graph is minimum over all such graphs. Such a graph is called an optimal realization and finding such realizations is known to be NP-hard. Recently Varone presented a heuristic greedy algorithm for computing optimal realizations. Here we present an alternative heuristic that exploits the relationship between realizations of the metric $D$ and its so-called tight span $T_D$. The tight span $T_D$ is a canonical polytopal complex that can be associated to $D$, and our approach explores parts of $T_D$ for realizations in a way that is similar to the classical simplex algorithm. We also provide computational results illustrating the performance of our approach for different types of metrics, including $l_1$-distances and two-decomposable metrics for which it is provably possible to find optimal realizations in their tight spans.
Let $mathfrak{M}$ be a class of metric spaces. A metric space $Y$ is minimal $mathfrak{M}$-universal if every $Xinmathfrak{M}$ can be isometrically embedded in $Y$ but there are no proper subsets of $Y$ satisfying this property. We find conditions under which, for given metric space $X$, there is a class $mathfrak{M}$ of metric spaces such that $X$ is minimal $mathfrak{M}$-universal. We generalize the notion of minimal $mathfrak{M}$-universal metric space to notion of minimal $mathfrak{M}$-universal class of metric spaces and prove the uniqueness, up to an isomorphism, for these classes. The necessary and sufficient conditions under which the disjoint union of the metric spaces belonging to a class $mathfrak{M}$ is minimal $mathfrak{M}$-universal are found. Examples of minimal universal metric spaces are constructed for the classes of the three-point metric spaces and $n$-dimensional normed spaces. Moreover minimal universal metric spaces are found for some subclasses of the class of metric spaces $X$ which possesses the following property. Among every three distinct points of $X$ there is one point lying between the other two points.
We establish sharp estimates for the $p$-capacity of metric rings with unrelated radii in metric measure spaces equipped with a doubling measure and supporting a Poincare inequality. These estimates play an essential role in the study of the local behavior of p-harmonic Greens functions.
We develop a combinatorial rigidity theory for symmetric bar-joint frameworks in a general finite dimensional normed space. In the case of rotational symmetry, matroidal Maxwell-type sparsity counts are identified for a large class of $d$-dimensional normed spaces (including all $ell^p$ spaces with $p ot=2$). Complete combinatorial characterisations are obtained for half-turn rotation in the $ell^1$ and $ell^infty$-plane. As a key tool, a new Henneberg-type inductive construction is developed for the matroidal class of $(2,2,0)$-gain-tight graphs.
In this note we prove that on general metric measure spaces the perimeter is equal to the relaxation of the Minkowski content w.r.t. convergence in measure
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