Abstract
In classical linear algebra, the question to know if a vector v ∈ ℝn belongs to the linear space V ect{v1, v2, ⋯ , vk} generated by a familly of vectors, is solved by the Gauss pivot. The problem investigated in this paper is very close to this classical question: we denote \( \left\lfloor \cdot \right\rfloor _n \) the function of ℝn defined by \( \left\lfloor {\left( {x_i } \right)_{1 \leqslant i \leqslant n} } \right\rfloor _n = \left( {\left\lfloor {x_i } \right\rfloor } \right)_{1 \leqslant i \leqslant n} \) and the question is now to determine if a given vector v ∈ ℤn belongs to \( \left\lfloor {Vect\left\{ {v^1 ,v^2 , \cdots ,v^k } \right\}} \right\rfloor _n \). This problem can be easily seen as a sytem of inequalities and solved by using linear programming but in some special cases, it can also be seen as a particular geometrical problem and solved by using tools of convex geometry. We will see in this framework that the question v ∈ \( \left\lfloor {Vect\left\{ {v^1 ,v^2 , \cdots ,v^k } \right\}} \right\rfloor _n \)? generalizes the problem of recognition of the finite parts of digital hyperplanes and we will give equivalent formulations which allow to solve it efficiently.
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Gérard, Y. (2000). A Question of Digital Linear Algebra. In: Borgefors, G., Nyström, I., di Baja, G.S. (eds) Discrete Geometry for Computer Imagery. DGCI 2000. Lecture Notes in Computer Science, vol 1953. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44438-6_11
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DOI: https://doi.org/10.1007/3-540-44438-6_11
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