Polygonal equalities and virtual degeneracy in $L_p$-spaces

Abstract

Suppose $0<p\le 2$ and that $(\Omega ,\mu )$ is a measure space for which $L_p(\Omega ,\mu )$ is at least two-dimensional. The central results of this paper provide a complete description of the subsets of $L_p(\Omega ,\mu )$ that have strict $p$-negative type. In order to do this we study non-trivial $p$-polygonal equalities in $L_p(\Omega ,\mu )$. 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 $\{z_1,\dots ,z_n\}$ is a subset of $L_p(\Omega ,\mu )$ and ${\alpha }_1,\dots ,{\alpha }_n$ are non-zero real numbers that sum to zero. We provide a complete classification of the non-trivial $p$-polygonal equalities in $L_p(\Omega ,\mu )$. 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 $L_p(\Omega ,\mu )$ 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 $q\ge p$, then: (1) $(X,d)$ is not isometric to any linear subspace $W$ of $L_p(\Omega ,\mu )$ that contains a pair of disjointly supported non-zero vectors, and (2) $(X,d)$ is not isometric to any subset of $L_p(\Omega ,\mu )$ 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 $L_2(\Omega ,\mu )$.

18 pages. This paper has been substantially revised and now includes all of the results from the withdrawn posting arXiv:1209.3454

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