Detecting and reconstructing centrally symmetric sets from the autocorrelation: Two discrete cases

Let A be a finite, centrally symmetric set in R d , d ≥ 1 , and let B be a set homometric to A , that is, for all x ∈ R d the sets A ∩ ( A + x ) and B ∩ ( B + x ) have equal cardinalities. We show that B is a translate of A . As a consequence, an analogous statement is obtained for bodies which are unions of lattice cubes. We notice that homometric sets are relevant in several research areas including image analysis, geometric probability, and X-ray crystallography.