Polygonal equalities and virtual degeneracy in [formula omitted]-spaces

Abstract

Suppose 0<p⩽2 and that (Ω,μ) is a measure space for which Lp(Ω,μ) is at least two-dimensional. 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 form∑j,i=1nαjαi‖zj−zi‖pp=0 where {z1,…,zn} is a subset of Lp(Ω,μ) and α1,…,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 L2(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 p1-polygonal equality for some p1∈[℘,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.