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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.

Enflo [3] constructed a countable metric space that may not be uniformly embedded into any metric space of positive generalized roundness. Dranishnikov, Gong, Lafforgue and Yu [2] modified Enflo's example to construct a locally finite metric space that may not be coarsely embedded into any Hilbert space. In this paper we meld these two examples into one simpler construction. The outcome is a locally finite metric space (Z, ζ) which is strongly non embeddable in the sense that it may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Moreover, we show that both types of embedding may be obstructed by a common recursive principle. It follows from our construction that any metric space which is Lipschitz universal for all locally finite metric spaces may not be embedded uniformly or coarsely into any metric space of non zero generalized roundness. Our construction is then adapted to show that the group Zω = ℵ 0 Z admits a Cayley graph which may not be coarsely embedded into any metric space of non zero generalized roundness. Finally, for each p ≥ 0 and each locally finite metric space (Z, d), we prove the existence of a Lipschitz injection f : Z → ℓp.2000 Mathematics Subject Classification. 46C05, 46T99.