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Register a new userComparing the generalized roundness of metric spaces
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Motivated by the local theory of Banach spaces we introduce a notion of finite representability for metric spaces. This allows us to develop a new technique for comparing the generalized roundness of metric spaces. We illustrate this technique in two different ways by applying it to Banach spaces and metric trees. In the realm of Banach spaces we obtain results such as the following: (1) if $mathcal{U}$ is any ultrafilter and $X$ is any Banach space, then the second dual $X^{astast}$ and the ultrapower $(X)_{mathcal{U}}$ have the same generalized roundness as $X$, and (2) no Banach space of positive generalized roundness is uniformly homeomorphic to $c_{0}$ or $ell_{p}$, $2 < p < infty$. Our technique also leads to the identification of new classes of metric trees of generalized roundness one. In particular, we give the first examples of metric trees of generalized roundness one that have finite diameter. These results on metric trees provide a natural sequel to a paper of Caffarelli, Doust and Weston. In addition, we show that metric trees of generalized roundness one possess special Euclidean embedding properties that distinguish them from all other metric trees.
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Every finite metric tree has generalized roundness strictly greater than one. On the other hand, some countable metric trees have generalized roundness precisely one. The purpose of this paper is to identify some large classes of countable metric trees that have generalized roundness precisely one. At the outset we consider spherically symmetric trees endowed with the usual combinatorial metric (SSTs). Using a simple geometric argument we show how to determine decent upper bounds on the generalized roundness of finite SSTs that depend only on the downward degree sequence of the tree in question. By considering limits it follows that if the downward degree sequence $(d_{0}, d_{1}, d_{2}...)$ of a SST $(T,rho)$ satisfies $|{j , | , d_{j} > 1 }| = aleph_{0}$, then $(T,rho)$ has generalized roundness one. Included among the trees that satisfy this condition are all complete $n$-ary trees of depth $infty$ ($n geq 2$), all $k$-regular trees ($k geq 3$) and inductive limits of Cantor trees. The remainder of the paper deals with two classes of countable metric trees of generalized roundness one whose members are not, in general, spherically symmetric. The first such class of trees are merely required to spread out at a sufficient rate (with a restriction on the number of leaves) and the second such class of trees resemble infinite combs.
We obtain several new characterizations of ultrametric spaces in terms of roundness, generalized roundness, strict p-negative type, and p-polygonal equalities (p > 0). This allows new insight into the isometric embedding of ultrametric spaces into Euclidean spaces. We also consider roundness properties additive metric spaces which are not ultrametric.
Enflo constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu modified Enflos example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space $(mathfrak{Z}, zeta)$ which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group $mathbb{Z}_omega=bigoplus_{aleph_0}mathbb{Z}$ admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each $p geq 0$ and each locally finite metric space $(Z,d)$, we prove the existence of a Lipschitz injection $f : Z to ell_{p}$.
In this article, the author proposes another way to define the completion of a metric space, which is different from the classical one via the dense property, and prove the equivalence between two definitions. This definition is based on considerations from category theory, and can be generalized to arbitrary categories.
In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardys original inequality. We give examples obtaining new weighted Hardy inequalities on $mathbb R^n$, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds.
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