Let $mathcal{M}(Omega, mu)$ denote the algebra of all scalar-valued measurable functions on a measure space $(Omega, mu)$. Let $B subset mathcal{M}(Omega, mu)$ be a set of finitely supported measurable functions such that the essential range of each
$f in B$ is a subset of ${ 0,1 }$. The main result of this paper shows that for any $p in (0, infty)$, $B$ has strict $p$-negative type when viewed as a metric subspace of $L_{p}(Omega, mu)$ if and only if $B$ is an affinely independent subset of $mathcal{M}(Omega, mu)$ (when $mathcal{M}(Omega, mu)$ is considered as a real vector space). It follows that every two-valued (Schauder) basis of $L_{p}(Omega, mu)$ has strict $p$-negative type. For instance, for each $p in (0, infty)$, the system of Walsh functions in $L_{p}[0,1]$ is seen to have strict $p$-negative type. The techniques developed in this paper also provide a systematic way to construct, for any $p in (2, infty)$, subsets of $L_{p}(Omega, mu)$ that have $p$-negative type but not $q$-negative type for any $q > p$. Such sets preclude the existence of certain types of isometry into $L_{p}$-spaces.
Negative type inequalities arise in the study of embedding properties of metric spaces, but they often reduce to intractable combinatorial problems. In this paper we study more quantitati
This work has been expanded and fully incorporated into arXiv:1203.5837. Cases of equality in the classical 2-negative type inequalities for Hilbert spaces are characterized in terms of balanced signed simplices. It follows that a metric subspace o
f a Hilbert space H has strict 2-negative type if and only if it is affinely independent (when H is considered as a real vector space). This allows a complete description of Shkarins class M.
Suppose $0 < p leq 2$ and that $(Omega, mu)$ is a measure space for which $L_{p}(Omega, mu)$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_{p}(Omega, mu)$ that have strict $p$-negat
ive type. In order to do this we study non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form begin{eqnarray*} sumlimits_{j, i = 1}^{n} alpha_{j} alpha_{i} {| z_{j} - z_{i} |}_{p}^{p} & = & 0 end{eqnarray*} where ${ z_{1}, ldots, z_{n} }$ is a subset of $L_{p}(Omega, mu)$ and $alpha_{1}, ldots, alpha_{n}$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$. The cases $p < 2$ and $p = 2$ are substantially different and are treated separately. The case $p = 1$ generalizes an elegant result of Elsner, Han, Koltracht, Neumann and Zippin. Another reason for studying non-trivial $p$-polygonal equalities in $L_{p}(Omega, mu)$ is due to the fact that they preclude the existence of certain types of isometry. For example, our techniques show that if $(X,d)$ is a metric space that has strict $q$-negative type for some $q geq p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_{p}(Omega, mu)$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_{p}(Omega, mu)$ that has non-empty interior. Furthermore, in the case $p = 2$, it also follows that $(X,d)$ is not isometric to any affinely dependent subset of $L_{2}(Omega, mu)$.
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 Eu
clidean 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 no
t 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 $f : Z to ell_{p}$.
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 tre
es 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.
