Vol. 76, No. 2, 1978

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On arc length sharpenings

William Andrew Ettling

Vol. 76 (1978), No. 2, 361–370

This paper introduces two new sharpenings:

Theorem. Let A denote a rectifiable arc (with length l(A)) of a metric space, let P denote a finite, normally-ordered subset of A, and let l(T(P)) denote the linear content of a mini-tree T(P) spanning P. Then l.u.b.PAl(T(P)) = l(A).

Definition. If E is a nonempty subset of a set P that is spanned by tree T, then T is said to be on E.

Theorem. Let σ(E) denote the greatest lower bound of the linear contents of all trees on E. If A denotes a rectifiable arc of a finitely compact metric space, then l.u.b.EAσ(E) = l(A), where E denotes any finite normally-ordered subset of A.

Mathematical Subject Classification 2000
Primary: 53C70
Received: 20 December 1976
Revised: 2 December 1977
Published: 1 June 1978
William Andrew Ettling