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The double midset conjecture for continua in the plane. (English) Zbl 0735.54020
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 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 A. D. Berard jun. and 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.
Reviewer: D.E.Bennett

54F15 Continua and generalizations
54D05 Connected and locally connected spaces (general aspects)
51M05 Euclidean geometries (general) and generalizations
51K05 General theory of distance geometry
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