An infinitesimal version of the Besicovitch-Danzer characterization of the circle (Q1108583)

\textit{A. S. Besicovitch} [J. Lond. Math. Soc. 36, 241-244 (1961; Zbl 0097.382)] and \textit{L. W. Danzer} [Convexity, Proc. Symp. Pure Math. 7, 99-100 (1963; Zbl 0135.225)] proved that every planar convex curve such that no rectangle has exactly three vertices on it is a circle. The author extends this characterization of the circle to Jordan curves. He introduces an infinitesimal version of the rectangle property by considering only rectangles with sidelengths ratio smaller than some fixed \(\epsilon >0\). Jordan curves satisfying this property are shown to be convex and of constant width. Furthermore they must be a circle if analyticity is assumed. The question if the infinitesimal rectangle property characterizes the circle remains open in the nonanalytic case. [As shown by the reviewer (``On the rectangle property for plane continua and immersed topological hypersurfaces'', Preprint, FB Mathematik, TU Berlin (1989)) the Besicovitch-Danzer theorem can be extended to plane continua. Also hyperspheres are characterized among topologically immersed hypersurfaces by the rectangle property.]