Arcwise Convex Sets

1. The concept of arcwise convex set is a natural generalization of the notion of L set. The latter was studied by Horn and Valentine [2]1. It is especially interesting to observe how the theorem about the complement of an arcwise convex continuum sheds light on the corresponding theorem for L sets. In order to make this precise, the following definitions are used. Definition 1. An arc C is said to be convex if it is contained in the boundary of its convex hull, denoted by 77(G). Definition 2. A set S is said to be arcwise convex if each pair of points in S can be joined by a convex arc in S. Definition 3. A continuum S in the Euclidean plane is called a unilaterally connected continuum if each pair of points x and y in S lies in a subcontinuum of S which is contained in one of the closed halfplanes determined by the line passing through x and y. An L set is an arcwise convex set, since, by definition, each pair of points in L can be joined by a polygonal line in L containing at most two segments. Notation 1. We let xy denote the closed linear segment joining x and y, and L(x, y) denotes the straight line passing through x and y. The convex hull of C is denoted by 77(C). The two open half-planes determined by a line 7,(x, y) in a plane are denoted by R+(x, y) and R~(x, y). A component of the complement of S is called K. In this paper we restrict ourselves to sets in a two-dimensional Euclidean space 7i2. The principal theorem in this section is: