Fundamenta Informaticae 146 (2016) 185–195 185 DOI 10.3233/FI-2016-1380 IOS Press On J-additivity and Bounded Additivity Sara Brunetti Dipartimento di Ingegneria dell’Informazione e Scienze Matematiche Via Roma, 56, 53100 Siena - Italy sara.brunetti@unisi.it Carla Peri ∗† Università Cattolica S. C. Via Emilia Parmense, 84, 29122 Piacenza - Italy carla.peri@unicatt.it Abstract. In this paper we consider the uniqueness issues in Discrete Tomography. A special class of geometric objects, widely considered in the literature, is represented by additive sets. These sets are uniquely determined by their X-rays, and they are also reconstructible in polyno- mial time by use of linear programming. Recently, additivity has been extended to J -additivity to provide a more general treatment of known concepts and results. A further generalization of additivity, called bounded additivity is obtained by restricting to sets contained in a given orthog- onal box. In this work, we investigate these two generalizations from a geometrical point of view and analyze the interplay between them. Keywords: Additive set, weakly bad configuration, uniqueness problem, X-ray 1. Introduction Discrete Tomography deals with reconstruction and uniqueness issues of a finite set of points F in Z n by the knowledge of its X -rays taken in a finite number of lattice directions. A vector u = (a 1 ,...,a n ) ∈ Z n , where a 1 ≥ 0, is said to be a lattice direction, if gcd(a 1 , ..., a n )=1. We refer to a finite subset F of Z n as a lattice set, and we denote its cardinality by |F |. Thus, let L u be the set of lines parallel to a lattice direction u, the X -ray v F of F in the direction u counts the number of points in F on each line ℓ ∈L u , and we define v F (ℓ)= |F ∩ ℓ| . ∗ Address for correspondence: Università Cattolica S. C., Via Emilia Parmense, 84, 29122 Piacenza, Italy. † Networking support was provided by the EXTREMA COST Action MP1207.