Polygonal equalities and virtual degeneracy in $L_{p}$-spaces

Details Polygonal equalities and virtual degeneracy in $L_{p}$-spaces Polygonal equalities and virtual degeneracy in $L_{p}$-spaces

2012-03-26

arxiv.org/abs/1203.5837v2

Mathematics

Functional Analysis

9 pages

Scientific paper

Cases of equality in the classical $p$-negative type inequalities for $L_{p}(\mu)$-spaces are characterized for each $p \in (0,2)$ according to a new property called virtual degeneracy. For each $p \in (0,2)$, this leads to a complete classification of the subsets of $L_{p}$-spaces that have strict $p$-negative type. It follows that if $0 < p < q \leq 2$ and if $(\Omega_{1}, \mu_{1})$ and $(\Omega_{2}, \mu_{2})$ are measure spaces, then no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any linear subspace $W$ of $L_{p}(\Omega_{1}, \mu_{1})$ that contains a pair of disjointly supported unit vectors. Under these circumstances it is also the case that no subset of $L_{q}(\Omega_{2}, \mu_{2})$ is isometric to any subset of $L_{p}(\Omega_{1}, \mu_{1})$ that has non-empty interior. We conclude the paper by examining virtually degenerate subspaces of $L_{p}(\mu)$-spaces.

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