Roundness properties of ultrametric spaces (Q2921056)

The paper is devoted to the geometry of metric spaces of some special types. The main classes of considered metrics are: ultrametrics, tree metrics (also known as additive metrics) and leaf metrics (leaf metrics are defined as restrictions of metrics of metric trees to their sets of leaves).NEWLINENEWLINEThe geometric properties studied in the paper are (in a sense) centered around the notion of \(p\)-negative type defined as follows: A metric space \((X,d)\) has \(p\)-negative type if and only if for all finite subsets \(\{x_{1}, \dots , x_{n} \} \subseteq X\) and all choices of real numbers \(\eta_{1}, \dots, \eta_{n}\) with \(\eta_{1} + \cdots + \eta_{n} = 0\), the condition \(\sum_{1 \leq i,j \leq n} d(x_{i},x_{j})^{p} \eta_{i} \eta_{j} \leq 0\) holds. The paper contains numerous results on geometric properties of the listed classes of metric spaces.NEWLINENEWLINE The authors' abstract describes some of the most important of them: ``Motivated by a classical theorem of Schoenberg we prove that an \(n + 1\) point finite metric space has strict \(2\)-negative type if and only if it can be isometrically embedded in the Euclidean space \(\mathbb{R}^{n}\) of dimension \(n\) but it cannot be isometrically embedded in any Euclidean space \(\mathbb{R}^{r}\) of dimension \(r < n\). We use this result as a technical tool to study `roundness' properties of additive metrics with a particular focus on ultrametrics and leaf metrics. The following conditions are shown to be equivalent for a metric space \((X,d)\): (1) \(X\) is ultrametric, (2) \(X\) has infinite roundness, (3) \(X\) has infinite generalized roundness, (4) \(X\) has strict \(p\)-negative type for all \(p \geq 0\), and (5) \(X\) admits no \(p\)-polygonal equality for any \(p \geq 0\). As all ultrametric spaces have strict \(2\)-negative type by (4) we thus obtain a short new proof of Lemin's theorem: Every finite ultrametric space is isometrically embeddable into some Euclidean space as an affinely independent set. Motivated by a question of Lemin, Shkarin introduced the class \(\mathcal{M}\) of all finite metric spaces that may be isometrically embedded into \(\ell_{2}\) as an affinely independent set. The results of this paper show that Shkarin's class \(\mathcal{M}\) consists of all finite metric spaces of strict \(2\)-negative type. We also note that it is possible to construct an additive metric space whose generalized roundness is exactly \(\wp\) for each \(\wp \in [1, \infty]\).''