Additivity obstructions for integral matrices and pyramids

In Discrete Tomography there are two related notions of interest: H-uniqueness and H-additivity of finite subsets of Nm, which are defined for certain finite sets H of linear subspaces of Rm. One knows complete sets of obstructions for H-uniqueness (bad H-configurations) and for H-additivity (weakly bad H-configurations). The classical case, when H is the set of coordinate axes in R2, is well known. Let Hm denote the set of the m coordinate hyperplanes of Rm. The following question was raised in [P.C. Fishburn, J.C. Lagarias, J.A. Reeds, L.A. Shepp, Sets uniquely determined by projections on axes II. Discrete case, Discrete Math. 91 (1991) 149–159]. Is there an upper bound on the weights of the bad Hm-configurations one needs to consider to determine Hm-uniqueness (m≥3) of an arbitrary set in Nm? This question can be asked for other sets H of linear subspaces and also for H-additivity. The answer to this question, in the case of uniqueness, is known when H is a set of lines. In this paper we answer this question for uniqueness and additivity in the case of H3. We show that there is no upper bound on the weights of the bad configurations (resp. weakly bad configurations) one needs to consider to determine H3-uniqueness (resp. H3-additivity).