#### RIS ID

29328

#### Abstract

Let (*X*,*d*) be a compact metric space and let (*X*) denote the space of all finite signed Borel measures on *X*. Define *I*:(*X*)→ by ...

and set *M*(*X*)=sup *I*(*μ*), where *μ* ranges over the collection of signed measures in (*X*) of total mass 1.

The metric space (*X*,*d*) is *quasihypermetric* if for all *n*, all *α*_{1},…,*α*_{n} satisfying ∑ _{i=1}^{n}*α*_{i}=0 and all *x*_{1},…,*x*_{n}*X*, the inequality ∑ _{i,j=1}^{n}*α*_{i}*α*_{j}*d*(*x*_{i},*x*_{j})≤0 holds. Without the quasihypermetric property *M*(*X*) is infinite, while with the property a natural semi-inner product structure becomes available on _{0}(*X*), the subspace of (*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 _{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 _{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 _{0}(*X*); and the functional-analytic properties of _{0}(*X*) as a semi-inner product space, including the question of its completeness. A later paper [P. Nickolas and R. Wolf, Distance geometry in quasihypermetric spaces. II, *Math. Nachr.*, accepted] will apply the work of this paper to a detailed analysis of the constant *M*(*X*).

## Publication Details

Nickolas, P. & Wolf, R. (2009). Distance geometry in quasihypermetric spaces. I. Bulletin of the Australian Mathematical Society, 80 (1), 1-25.