Vol. 56, No. 2, 1975

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Generalized sums of distances

Ralph Alexander

Vol. 56 (1975), No. 2, 297–304

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)(x)(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
Received: 13 December 1973
Revised: 17 September 1974
Published: 1 February 1975
Ralph Alexander