an:00040181
Zbl 0754.54025
Loveland, L. D.; Loveland, S. M.
Equidistant sets in plane triodic continua
EN
Proc. Am. Math. Soc. 115, No. 2, 553-562 (1992).
0002-9939 1088-6826
1992
j
54F15 54F50 51K05 54F65
bisector; triod; midset property; simple closed curve; arc
For each positive integer \(n\), a metric space \(X\) is said to have the \(n\)-point midset property (shortly \(n\)-MP) if for every two points \(x\) and \(y\) in \(X\) the set of all points of \(X\) equidistant from \(x\) and \(y\) consists of \(n\) points. Generalizing earlier results, the main theorem of the paper states that if a continuum in the Euclidean plane has the \(n\)- MP for \(n\geq 1\), then it must either be a simple closed curve or an arc. It is remarked at the end of the paper that in a forthcoming paper the authors have proved even a stronger result: if a planar continuum \(X\) has the \(n\)-MP for \(n\geq 1\), then either \(n=1\) and \(X\) is an arc, or \(n=2\) and \(X\) is a simple closed curve.
J.J.Charatonik (Wrocław)