Type: Article

Additivity obstructions for integral matrices and pyramids

Miguel Santoyo; Ernesto Vallejo;

Abstract:

In Discrete Tomography there are two related notions of interest: H-uniqueness and H-additivity of finite subsets of N-m, which are defined for certain finite sets of linear subspaces of R-m. 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 R-2, is well known. Let H-m 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, 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 X.-configurations one needs to consider to determine H-m-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 H-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-3-uniqueness (resp. H-3-additivity)