Sets determined by finitely many X-rays

✍ Scribed by R. J. Gardner

• Publisher: Springer
• Year: 1992
• Tongue: English
• Weight: 782 KB
• Volume: 43
• Category: Article
• ISSN: 0046-5755
• DOI: 10.1007/bf00181293

✦ Synopsis

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 Y of directions, or more generally by its X-rays parallel to a finite set Y of subspaces, is called Y-unique, or simply unique. Some subclasses of the Y-unique sets are known. The Y-additive sets are those measurable sets E which can be written E ~ {x E Rn:Y.lf/(x) * 0}. Here, ~ denotes equality modulo null sets, * is either /> or >, and the terms in the sum are the values of ridge functionsf~ orthogonal to subspaces Si in Y. If n = 2, the Y-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 :7. Relations between these classes are investigated. It is shown that in R 2 each Y-inscribable convex set is Y-additive, but Y-additive convex sets need not be 5¢-inscribable. It is also shown that every ellipsoid in En is unique for any set of three directions. Finally, some results are proved concerning the structure of convex sets in g~n, unique with respect to certain families of coordinate subspaces.