DISTANCE GEOMETRY IN QUASIHYPERMETRIC SPACES. I

PETER NICKOLAS, REINHARD WOLF
2009 Bulletin of the Australian Mathematical Society  
Let (X, d) be a compact metric space and let M(X ) denote the space of all finite signed Borel measures on X . Define I : M(X ) → R by and set M(X ) = sup I (µ), where µ ranges over the collection of signed measures in M(X ) of total mass 1. The metric space (X, d) is quasihypermetric if for all n ∈ N, all α 1 , . . . , α n ∈ R satisfying n i=1 α i = 0 and all x 1 , . . . , x n ∈ X , the inequality n i, j=1 α i α j d(x i , x j ) ≤ 0 holds. Without the quasihypermetric property M(X ) is
more » ... while with the property a natural semi-inner product structure becomes available on M 0 (X ), the subspace of 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 semiinner product space structure of 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 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 M 0 (X ); and the functional-analytic properties of M 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 ). 2000 Mathematics subject classification: primary 51K05; secondary 54E45, 31C45.
doi:10.1017/s0004972708000932 fatcat:ncyso4dmwrfzblyboj5apu3tvm