INVERSION OF THE CROFTON TRANSFORM FOR SETS IN THE PLANE

Abstract

The genesis of this paper goes back to a question posed years ago by the late H. Steinhaus, “About two plane arcs both of finite lengths it is known that every line in the plane meets both at the same number of points (which may be zero or $\infty$). Must the arcs be identical ?” This question in answered in this paper using the Crofton transform. For a fixed set in the plane the Crofton transform for that set is defined as the number of points in which the set meets a variable line in the plane. In this paper we construct an inversion of the Crofton transform within a certain class of plane Borel sets, a somewhat weak form of inversion retrieving from the Crofton transform (actually from the cross-integral function, which is somewhat stronger) the closure of the set rather than the set (or the set modulo a set of linear measure zero) itself. We also establish uniqueness (or rather a degree of uniqueness) of two plane sets from the same class, whose Crofton transforms coincide over certain families of lines. This answers a stronger version of the original Steinhaus question.