Classification of the convexity properties

Abstract

During the last ten years a tendency of “organising” the wide range of convexity properties for sets and for functions appeared in the mathematical literature. For example, the great collection of P. M. Gruber and J. M. Willis (1993) refers to 21 types of convexity for various types of sets. I. Singer (1997) writes about 26 types of convexity for sets, trying to obtain a classification of them according to the principle of definition and of some properties. The thesis of J. M. Chassery (1984 [43]) refers to 6 types of discrete convexity properties. Our own book, G. Cristescu (2000 [71]), contains 19 notions of convexity for sets, trying to build a unifying theory of most of them. Other references and studies on more types of convexities or concavities for sets are in the books published or coordinated by S. Schaible (1981), M. Avriel et al. (1988) and V.P. Soltan (1984).