Polygonal equalities and virtual degeneracy in -spaces

Suppose 0 < p ⩽ 2 and that ( Ω , μ ) is a measure space for which L p ( Ω , μ ) is at least two-dimensional. The central results of this paper provide a complete description of the subsets of L p ( Ω , μ ) that have strict p-negative type. In order to do this we study non-trivial p-polygonal equalities in L p ( Ω , μ ) . These are equalities that can, after appropriate rearrangement and simplification, be expressed in the form ∑ j , i = 1 n α j α i ‖ z j − z i ‖ p p = 0 where { z 1 , … , z n } is a subset of L p ( Ω , μ ) and α 1 , … , α 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 ( 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 L p ( 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 L p ( Ω , μ ) that contains a pair of disjointly supported non-zero vectors, and (2) ( X , d ) is not isometric to any subset of L p ( 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 ℓ p that do not have strict p-negative type.