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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

##### MSC:
 54F15 Continua and generalizations 54D05 Connected and locally connected spaces (general aspects) 51M05 Euclidean geometries (general) and generalizations 51K05 General theory of distance geometry
##### Keywords:
bisector; double midset property; simple closed curve
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##### References:
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