Polygonal equalities and virtual degeneracy in $L_{p}$-spaces
Suppose $0 < p \leq 2$ 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 \begin{eqnarray*} \sum\limits_{j, i = 1}^{n} α_{j} α_{i} {\| z_{j} - z_{i} \|}_{p}^{p} & = & 0 \end{eqnarray*} where $\{ z_{1}, \ldots, z_{n} \}$ is a subset of $L_{p}(Ω, μ)$ and $α_{1}, \ldots, α_{n}$ 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 $q \geq p$, 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 $L_{2}(Ω, μ)$.