On the rectangle property for plane continua and immersed topological hypersurfaces (Q1122784)

\textit{A. S. Besicovitch} [J. Lond. Math. Soc. 36, 241--244 (1961; Zbl 0097.38201)] and \textit{L. Danzer} [Proc. Symp. Pure Math. 7, 99--100 (1963; Zbl 0135.22503)] proved the following characterization of the circle: Every planar convex closed curve such that no rectangle has exactly three vertices on it is a circle. This characterization has been extended by \textit{T. Zamfirescu} [Geom. Dedicata 27, 209--212 (1988; Zbl 0654.52001)] to closed Jordan curves. There also an infinitesimal version of this so-called rectangle property has been established characterizing convex curves of constant width among closed Jordan curves. In the analytic case they only can be circles. Here the open problem (proposed by T. Zamfirescu) of extending this characterization to plane continua is solved. The rectangle property only can be valid for circles and special kinds of Jordan arcs. Clearly the proof is rather complicated because plane continua are a comparatively general kind of spaces. Furthermore two other characterization results are presented in the plane case. In higher dimensions topologically immersed hypersurfaces with rectangle property are shown to be hyperspheres. But if the codimension is greater than one there are examples of closed \(C^{\infty}\) submanifolds of Euclidean space satisfying the rectangle property without being a sphere. This disproves a conjecture due to T. Zamfirescu [the paper cited above].