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This work has been expanded and fully incorporated into arXiv:1203.5837. Cases of equality in the classical 2-negative type inequalities for Hilbert spaces are characterized in terms of balanced signed simplices. It follows that a metric subspace of a Hilbert space H has strict 2-negative type if and only if it is affinely independent (when H is considered as a real vector space). This allows a complete description of Shkarins class M.
Suppose $0 < p leq 2$ 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 $p$-negat
We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidist
A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we gener
Coherent states in a projected Hilbert space have many useful properties. When there are conserved quantities, a representation of the entire Hilbert space is not necessary. The same issue arises when conditional observations are made with post-selec
We study the Wasserstein space (with quadratic cost) of Euclidean spaces as an intrinsic metric space. In particular we compute their isometry groups. Surprisingly, in the case of the line, there exists a (unique) exotic isometric flow. This contrast