Additivity obstructions for integral matrices and pyramids

摘要

In Discrete Tomography there are two related notions of interest: H-uniqueness and H-additivity of finite subsets of N{sup}m, which are defined for certain finite sets H of linear subspaces of R{sup}m. One knows complete sets of obstructions for H-uniqueness (bad reconfigurations) and for H-additivity (weakly bad H-configurations). The classical case, when H is the set of coordinate axes in R{sup}2, is well known. Let H{sub}m denote the set of the m coordinate hyperplanes of R{sup}m. The following question was raised in [P.C. Fishburn, J.C. Lagarias, J.A. Reeds, LA 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 H{sub}m-configurations one needs to consider to determine H{sub}m-uniqueness (m ≥ 3) of an arbitrary set in N{sup}m? This question can be asked for other sets H of linear subspaces and also for H{sub}-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 H{sub}3. 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 H{sub}3-uniqueness (resp. H{sub}3-additivity).