Document Type

Article

Publication Date

6-1997

Abstract

We study the determination of finite subsets of the integer lattice Zn, n ≥ 2, by X-rays. In this context, an X-ray of a set in a direction u gives the number of points in the set on each line parallel to u. For practical reasons, only X-rays in lattice directions, that is, directions parallel to a nonzero vector in the lattice, are permitted. By combining methods from algebraic number theory and convexity, we prove that there are Sour prescribed lattice directions such that convex subsets of Zn (i.e., finite subsets F with F = Zn ∩ conv F) are determined, among all such sets, by their X-rays in these directions. We also show that three X-rays do not suffice for this purpose. This answers a question of Larry Shepp, and yields a stability result related to Hammer's X-ray problem. We further show that any set of seven prescribed mutually nonparallel lattice directions in Z2 have the property that convex subsets of Z2 are determined, among all such sets, by their X-rays in these directions. We also consider the use of orthogonal projections in the interactive technique of successive determination, in which the information from previous projections can be used in deciding the direction for the next projection. We obtain results for finite subsets of the integer lattice and also for arbitrary finite subsets of Euclidean space which are the best possible with respect to the numbers of projections used.

Publication Title

Transactions of the American Mathematical Society

Volume

349

Issue

6

First Page

2271

Last Page

2295

Required Publisher's Statement

First published in The Transactions of the American Mathematical Society in Volume 349, Number 6, June 1997, published by the American Mathematical Society

Included in

Mathematics Commons

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