Convexities defined by means of distance functions
It was proved in J. M. Chassery (1983) that if a set A ⊂ Z2(h) is discrete convex (in the sense of definition 4.3.2) then there is a convex set inR2 such that A is its digitisation. In this chapter we intend to study the sets that are the solution of the converse problem: to find the properties of the sets that become, after digitisation, the same discrete convex set. The properties of convexity with respect to a set and two behaviours, studied in the previous chapter, will allow us to solve the above-mentioned problem. The methods of digitisation are not studied in this chapter. They are discussed in the thesis of M. L. P. van Lierop (1987) and further in M. Maes (1990). In chapter 10 we shall choose a particular digitisation method that leads to all the results of this chapter, but it is not unique, the results being easy to retrieve if the digitisation method is changed.
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