Polygonal equalities and virtual degeneracy inLp-spaces

Abstract

Suppose 0 < p ≤ 2 and that (Ω, µ) is a measure space for which Lp(Ω, µ) is at least twodimensional. The central results of this paper provide a complete description of the subsets of Lp(Ω, µ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in Lp(Ω, µ). These are equalities that can, after appropriate rearrangement and simplification, be expressed in the formwhere {z 1 , . . . , zn} is a subset of Lp(Ω, µ) and α 1 , . . . , αn are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial p-polygonal equalities in Lp(Ω, µ). 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 Lp(Ω, µ) 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 ≥ p, then: (1) (X, d) is not isometric to any linear subspace W of Lp(Ω, µ) that contains a pair of disjointly supported non-zero vectors, and (2) (X, d) is not isometric to any subset of Lp(Ω, µ) 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 (Ω, µ). More generally, we show that if (Y, ρ) is a metric space whose generalized roundness ℘ is finite and if (X, d) is a metric space that has strict q-negative type for some q ≥ ℘, then (X, d) is not isometric to any metric subspace of (Y, ρ) that admits a non-trivial p 1 -polygonal equality for some p 1 ∈ [℘, q]. It is notable in all of these statements that the metric space (X, d) can, for instance, be any ultrametric space. As a result we obtain new insights into sophisticated embedding theorems of Lemin and Shkarin.We conclude the paper by constructing some pathological infinite-dimensional linear subspaces of ℓp that do not have strict p-negative type.2010 Mathematics Subject Classification. 54E40, 46B04, 46C05.

Comparing the Generalised Roundness of Metric Spaces

“…By Lennard et al [25], this implies that (Z, d) has p-negative type for some p ∈ (1,2]. Consequently, the metric transform (T, √ d p ) embeds isometrically into 2 by Kelleher et al [20,Theorem 5.6]. We conclude that the only metric trees that satisfy condition (2) are those of generalised roundness one.…”

“…The proof of Theorem 5.2 shows that all metric trees have strict 1-negative type. Therefore, every metric tree satisfies Theorem 5.2(1) by the result of Kelleher et al [20].…”

“…Equivalently, the metric transform (T, √ d) has strict 2-negative type. Therefore, (T, √ d) is isometric to an affinely independent subset of 2 by Kelleher et al [20,Theorem 5.6]. This gives condition (1).…”

The geometry of two-valued subsets of Lp -spaces

“…This theorem underpins the main considerations of this paper. Theorem 2.3 (Kelleher et al [7]). Let n ≥ 1 be an integer and let X be a real or complex vector space.…”

“…For p ∈ (0, 2), Kelleher et al [7] have shown that if a subset B of L p (Ω, µ) is affinely independent (when L p (Ω, µ) is considered as a real vector space), then B has strict p-negative type. The converse of this statement is true when p = 2 but not when p < 2 (see Theorem 2.4 and Remark 1).…”