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

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 . 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  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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