On arc length sharpenings

William Andrew Ettling

Vol. 76 (1978), No. 2, 361–370

DOI: 10.2140/pjm.1978.76.361

Abstract

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._P⊂Al(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._E⊂Aσ(E) = l(A), where E denotes any finite normally-ordered subset of A.

Mathematical Subject Classification 2000

Primary: 53C70

Milestones

Received: 20 December 1976

Revised: 2 December 1977

Published: 1 June 1978