On congruence indices for simple closed curves

Abstract:

L. M. Blumenthal has defined the concept of congruence indices for point sets in his book Theory and applications of distance geometry, Clarendon Press, Oxford, 1953. Blumenthal shows that the circle has congruence indices (3, 1) and asks if this characterizes the circle among the class of simple closed curves. In this paper it is established that various classes of simple closed curves do not have congruence indices $(3,n)$ for any $n$. Included in these classes are the polygons, simple closed curves with convex interiors and a straight line segment contained in the curve, and simple closed curves with continuous nonconstant radius of curvature on some arc. Thus any noncircular simple closed curve with congruence indices (3, 1) must be very pathological. It is shown that if a simple closed curve has positive planar Lebesgue measure, then it fails to have congruence indices $(3,n)$ for any $n$.