Generalized sums of distances

Ralph Alexander

Vol. 56 (1975), No. 2, 297–304

DOI: 10.2140/pjm.1975.56.297

Abstract

Let K be a compact set in a Euclidean space and let d be a metric on K which is continuous with respect to the usual topology. The generalized energy integral I(μ) = ∫ ∫ d(x,y)dμ(x)dμ(y) is investigated as μ is allowed to range over the lamily of signed Borel measures of total mass one concentrated on K. A trick of integral geometry is used to define a class of metrics d, including many standard ones, possessing a number of pleasing properties related to the functional I.

Mathematical Subject Classification 2000

Primary: 52A20

Secondary: 53C65, 60D05

Milestones

Received: 13 December 1973

Revised: 17 September 1974

Published: 1 February 1975