an:00013929
Zbl 0735.54020
Loveland, L. D.
The double midset conjecture for continua in the plane
EN
Topology Appl. 40, No. 2, 117-129 (1991).
0166-8641
1991
j
54F15 54D05 51M05 51K05
bisector; double midset property; simple closed curve
A metric space \(X\) is said to have the double midset property (DMP) if the set of all points equidistant from any given two points of \(X\) consists of exactly two points. The author with \textit{S. G. Wayment} [Am. Math. Mon. 81, 1003-1006 (1974; Zbl 0291.54042)] conjectured that a continuum with DMP is a simple closed curve. Moreover \textit{A. D. Berard jun.} and \textit{W. Nitka} [Fundam. Math. 85, 49-55 (1974; Zbl 0281.53042)] conjectured that a nondegenerate, connected, metric space with the DMP is a simple closed curve. The author proves that if a continuum in the Euclidean plane has DMP, then it is a simple closed curve. The more general conjectures remain open.
D.E.Bennett
0291.54042; 0281.53042