Abstract. Consider the class of all measurable plane sets having given horizontal and vertical projections. According to this class the plane can be divided into three subsets: the essentially common subset of all elements of the class, the essentially common subset of the complements of all elements of the class, and the remaining subset of the plane. The three sets together are called the structure of the class. In this paper a method is given by which the structure of an arbitrary class can be determined from the projections. The method is similar to the procedure applied in the case of binary matrices. First, the structure of the normalized class (having rearranged non-increasing projections from the original ones) is constructed. Then, by a measure preserving mapping the structure of the original class is derived from the structure of the normalized class. The structure can be used in the reconstruction of non-unique sets from their projections, e.g. it gives information about the shape and the position of the possible solutions and an upper bound of the measure of the difference between two solutions.

AMS Subject Classification (1991): 28A05, 28A45

Keyword(s): projections, uniqueness, rearrangements, reconstruction

Received April 1, 1993. (Registered under 5553/2009.)