Generalizations of convexity

I consider four generalizations of the concept of a convex set in Rd. A subset X of R belongs to the family T(a) if for all x, y e X ax + (1 -a)y e X where a e R. Properties of elements of T(a) are considered in Chapter 1.Also in Chapter 1 a planar generalization of the family T(a) is considered. In Chapter 2 a study of m-convex sets is made and the extensive literature is constructively reviewed. In Chapter 3 some properties of locally starshaped sets are obtained.