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Recognition of Digital Polyhedra with a Fixed Number of Faces Is Decidable in Dimension 3

  • Yan GérardEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10502)

Abstract

We consider a conjecture on lattice polytopes \(Q\subset \mathbb {R}^d\) (the vertices are integer points) or equivalently on finite subsets \(S\subset \mathbb {Z}^d\), Q and S being related by \(Q\cap \mathbb {Z}^d = S\) or \(Q=\mathrm {conv (S)}\): given the vertices of Q or the list of points of S and an integer n, the problem to determine whether there exists a (rational) polyhedron \(P\subset \mathbb {R}^d\) with at most n faces and verifying \(P \cap \mathbb {Z}^d= S\) is decidable.

In terms of computational geometry, it’s a problem of polyhedral separability of S and \(\mathbb {Z}^d \setminus S\) but the infinite number of points of \(\mathbb {Z}^d \setminus S\) makes it intractable for classical algorithms. This problem of digital geometry is however very natural since it is a kind of converse of Integer Linear Programming.

The conjecture is proved in dimension \(d=2\) and in arbitrary dimension for non hollow lattice polytopes Q [6]. The purpose of the paper is to extend the result to hollow polytopes in dimension \(d=3\). An important part of the work is already done in [5] but it remains three special cases for which the set of outliers can not be reduced to a finite set: planar sets, pyramids and marquees. Each case is solved with a particular method which proves the conjecture in dimension \(d=3\).

Keywords

Pattern recognition Geometry of numbers Polyhedral separation Digital polyhedron Hollow lattice polytopes 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.LIMOS - UMR 6158 CNRS/Université Clermont AuvergneClermont-ferrandFrance

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