Sums, Projections, and Sections of Lattice Sets, and the Discrete Covariogram

Abstract

Abstract. Basic properties of finite subsets of the integer lattice Z n are investigated from the point of view of geometric tomography. Results obtained concern the Minkowski addition of convex lattice sets and polyominoes, discrete X-rays and the discrete and continuous covariogram, the determination of symmetric convex lattice sets from the cardinality of their projections on hyperplanes, and a discrete version of Meyer's inequality on sections of convex bodies by coordinate hyperplanes.

Some problems about geometric lattice

“…Gardner, Gronchi and Zong also asked in [1]: under what additional condition the discrete version of the Aleksandrov problem has an affirmative answer in …”

“…A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of  n u S , is 1 C included in 2 C by some suitable translation?…”

Konvexgeometrie

“…The paper [BG06] also constructs two different model sets with equal diffraction image. This construction uses as windows two nonconvex polygons with equal covariogram discovered in [GGZ05].…”

Homometric model sets and window covariograms

“…Other examples include convex sets in the plane with C 2 -smooth boundaries, and large other classes (e.g., non-C 1 bodies and bodies that are not strictly convex), though the general claim is still open, see [7]. However, starting with dimension 4, uniqueness fails even within the class of convex polytopes, compare [7,12]. In general, the reconstruction of K from the knowledge of g K is a difficult problem, which is beyond our scope here.…”

“…Similar questions emerge for non-empty compact subsets K ⊂ R d , where the covariogram g K (x) = vol K ∩ (x + K) encapsulates the difference information, and one tries to determine K from the knowledge of g K , see [7,12]. Furthermore, one is also interested in similar concepts for infinite point sets in general, and Delone sets in particular.…”