Uniqueness in discrete tomography of Delone sets with long-range order

Huck, Christian (2009). Uniqueness in discrete tomography of Delone sets with long-range order. Discrete and Computational Geometry, 42(4) pp. 740–758.

DOI: https://doi.org/10.1007/s00454-009-9213-z

Abstract

We address the problem of determining finite subsets of Delone sets Λ⊂ℝ^d with long-range order by X-rays in prescribed Λ-directions, i.e., directions parallel to nonzero interpoint vectors of Λ. Here, an X-ray in direction u of a finite set gives the number of points in the set on each line parallel to u. For our main result, we introduce the notion of algebraic Delone sets Λ⊂ℝ² and derive a sufficient condition for the determination of the convex subsets of these sets by X-rays in four prescribed Λ-directions.

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