On Sets which Meet each Line in Exactly Two Points

Using techniques from geometric measure theory and descriptive set theory, we prove a general result concerning sets in the plane which meet each straight line in exactly two points. As an application, we show that no such ‘two‐point’ set can be expressed as the union of countably many rectifiable sets together with a set with Hausdorff 1‐measure zero. Also, as a corollary, we show that no analytic set can be a two‐point set provided that every purely unrectifiable set meets some line in at least three points. Some generalizations are given to ‘ n ‐point’ sets and some other geometric constructions. 1991 Mathematics Subject Classification 28A05, 54C50.