TY - GEN A1 - Averkov, Gennadiy A1 - Langfeld, Barbara T1 - Homometry and direct-sum decompositions of lattice-convex sets T2 - Discrete & computational geometry : an international journal of mathematics and computer science N2 - Two sets in Rd are called homometric if they have the same covariogram, where the covariogram of a finite subset K of Rd is the function associating to each u∈Rd the cardinality of K∩(K+u). Understanding the structure of homometric sets is important for a number of areas of mathematics and applications. If two sets are homometric but do not coincide up to translations and point reflections, we call them nontrivially homometric. We study nontrivially homometric pairs of lattice-convex sets, where a set K is called lattice-convex with respect to a lattice M⊆Rd if K is the intersection of M and a convex subset of Rd. This line of research was initiated in 2005 by Daurat, Gérard and Nivat and, independently, by Gardner, Gronchi and Zong. All pairs of nontrivially homometric lattice-convex sets that have been known so far can essentially be written as direct sums S⊕T and S⊕(−T), where T is lattice-convex, the underlying lattice M is the direct sum of T and some sublattice L, and S is a subset of L. We study pairs of nontrivially homometric lattice-convex sets assuming this particular form and establish a necessary and a sufficient condition for the lattice-convexity of S⊕T. This allows us to explicitly describe all nontrivially homometric pairs in dimension two, under the above assumption, and to construct examples of nontrivially homometric pairs of lattice-convex sets for each d≥3. KW - Covariogram problem KW - Diffraction KW - Discrete tomography KW - Direct sum KW - Homometric sets KW - Lattice-convex set Y1 - 2016 UR - https://link.springer.com/article/10.1007%2Fs00454-016-9786-2 U6 - https://doi.org/10.1007/s00454-016-9786-2 SN - 1432-0444 SN - 0179-5376 VL - 56 IS - 1 SP - 216 EP - 249 ER - TY - GEN A1 - Averkov, Gennadiy A1 - Langfeld, Barbara T1 - On the Reconstruction of Planar Lattice-Convex Sets from the Covariogram T2 - Discrete & computational geometry N2 - A finite subset K of ℤ d is said to be lattice-convex if K is the intersection of ℤ d with a convex set. The covariogram g K of K⊆ℤ d is the function associating to each u∈ℤ d the cardinality of K∩(K+u). Daurat, Gérard, and Nivat and independently Gardner, Gronchi, and Zong raised the problem of the reconstruction of lattice-convex sets K from g K . We provide a partial positive answer to this problem by showing that for d=2 and under mild extra assumptions, g K determines K up to translations and reflections. As a complement to the theorem on reconstruction we also extend the known counterexamples (i.e., planar lattice-convex sets which are not reconstructible, up to translations and reflections) to an infinite family of counterexamples. KW - Covariogram KW - Crystallography KW - Diffraction KW - Direct sum KW - Homometric sets KW - Lattice-convex set KW - Matheron’s problem KW - Partial digest problem KW - Quasicrystal KW - Tomography KW - X-ray Y1 - 2012 UR - https://link.springer.com/article/10.1007%2Fs00454-012-9416-6 U6 - https://doi.org/10.1007/s00454-012-9416-6 SN - 1432-0444 VL - 48 IS - 1 SP - 216 EP - 238 ER -