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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.
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
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
We define a locally convex space $E$ to have the $Josefson$-$Nissenzweig$ $property$ (JNP) if the identity map $(E,sigma(E,E))to ( E,beta^ast(E,E))$ is not sequentially continuous. By the classical Josefson--Nissenzweig theorem, every infinite-dimens
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
Let $L$ be a fixed branch -- that is, an irreducible germ of curve -- on a normal surface singularity $X$. If $A,B$ are two other branches, define $u_L(A,B) := dfrac{(L cdot A) : (L cdot B)}{A cdot B}$, where $A cdot B$ denotes the intersection numbe