Article

# Distance Geometry in Quasihypermetric Spaces. III

Mathematische Nachrichten (Impact Factor: 0.58). 10/2008; DOI: 10.1002/mana.200810216

Source: arXiv

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**ABSTRACT:**For some centrally symmetric convex bodies $K\subset \mathbb R^n$, we study the energy integral $$ \sup \int_{K} \int_{K} \|x - y\|_r^{p}\, d\mu(x) d\mu(y), $$ where the supremum runs over all finite signed Borel measures $\mu$ on $K$ of total mass one. In the case where $K = B_q^n$, the unit ball of $\ell_q^n$ (for $1 \leq q \leq 2$) or an ellipsoid, we obtain the exact value or the correct asymptotical behavior of the supremum of these integrals. We apply these results to a classical embedding problem in metric geometry. We consider in $\mathbb R^n$ the Euclidean distance $d_2$. For $0 < \alpha < 1$, we estimate the minimum $R$ for which the snowflaked metric space $(K, d_2^{\alpha})$ may be isometrically embedded on the surface of a Hilbert sphere of radius $R$.12/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**Let $(X, d)$ be a compact metric space and let $\mathcal{M}(X)$ denote the space of all finite signed Borel measures on $X$. Define $I \colon \mathcal{M}(X) \to \R$ by \[I(\mu) = \int_X \int_X d(x,y) d\mu(x) d\mu(y),\] and set $M(X) = \sup I(\mu)$, where $\mu$ ranges over the collection of signed measures in $\mathcal{M}(X)$ of total mass 1. The metric space $(X, d)$ is quasihypermetric if for all $n \in \N$, all $\alpha_1, ..., \alpha_n \in \R$ satisfying $\sum_{i=1}^n \alpha_i = 0$ and all $x_1, ..., x_n \in X$, one has $\sum_{i,j=1}^n \alpha_i \alpha_j d(x_i, x_j) \leq 0$. Without the quasihypermetric property $M(X)$ is infinite, while with the property a natural semi-inner product structure becomes available on $\mathcal{M}_0(X)$, the subspace of $\mathcal{M}(X)$ of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of $(X, d)$, the semi-inner product space structure of $\mathcal{M}_0(X)$ and the Banach space $C(X)$ of continuous real-valued functions on $X$; conditions equivalent to the quasihypermetric property; the topological properties of $\mathcal{M}_0(X)$ with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-$*$ topology and the measure-norm topology on $\mathcal{M}_0(X)$; and the functional-analytic properties of $\mathcal{M}_0(X)$ as a semi-inner product space, including the question of its completeness. A later paper [Peter Nickolas and Reinhard Wolf, Distance Geometry in Quasihypermetric Spaces. II] will apply the work of this paper to a detailed analysis of the constant $M(X)$.Bulletin of the Australian Mathematical Society 10/2008; · 0.48 Impact Factor

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