Convexity with respect to a set and two behaviours

Abstract

Another way of obtaining new types of convexity properties for sets by means of the notion of behaviour defined in the chapter 3 consists in the combination of the techniques of superposition of a set on another one, more often a network over its support set, with the description of the relationship between their elements and their subsets using the above presented technique of behaviours. In this chapter we shall show how does the concept of behaviour allow the formulation of the definition of a convexity property of a set, which is general enough to contain, as particular cases, convexities intervening in various domains of mathematics. In the integer programming, the set of points of Z ⁿ is considered as a network over R ⁿ and the convexity in Z ⁿ appears as a restriction of a convex set from R ⁿ to Z ⁿ. It is the case of the results from L. Lup§a (1980). More general, if the support set is a complete lattice and the network over it is a sub-lattice then we obtain the convexity defined within the geometry of preference spaces by V. B. Kuzmin, S. V. Ovchinnikov (1975), S. V. Ovchinnikov (1980)). L. Lup§a (1986) has unified all these properties into a general theory. We shall also take into account the convexities appearing in image analysis. In this domain, the set of pixels is a network over R ² and the convexity of figures in the plane is perceived through this network. Various approaches of the convexity from this directions can be found in the papers of J. M. Chassery (1983, 1984), C.E. Kim, A. Rosenfeld (1980), C.E. Kim, J. Sklansky (1982), L. Latecki, A. Rosenfeld, R. Silverman (1955), J. F. Lawrence, W. R. Hare, J. M. Kenelly (1972)). G. Cristescu (1996 [56], [58], [59]) has studied these types of properties in a more general framework. The properties of convexity presented in this chapter do not imply, generally, the connectivity, as it will be shown by means of more examples.