A measurable set in R(n) which is uniquely determined among all measurable sets (modulo null sets) by its X-rays in a finite set J of directions, or more generally by its X-rays parallel to a finite set J of subspaces, is called J-unique, or simply unique. Some subclasses of the J-unique sets are known. The J-additive sets are those measurable sets E which can be written E approximately {x is-an-element-of R(n):SIGMA(i)f(i)(x) * 0}. Here, approximately denotes equality modulo null sets, * is either greater-than-or-equal-to or >, and the terms in the sum are the values of ridge functions f(i) orthogonal to subspaces S(i) in J. If n = 2, the J-inscribable convex sets are those whose interiors are the union of interiors of inscribed convex polygons, all of whose sides are parallel to the lines in J* Relations between these classes are investigated. It is shown that in R2 each J-inscribable convex set is J-additive, but J-additive convex sets need not be 9-inscribable. It is also shown that every ellipsoid in R(n) is unique for any set of three directions. Finally, some results are proved concerning the structure of convex sets in R(n), unique with respect to certain families of coordinate subspaces.
