Characterizations of Convex Sets by Local Support Properties

It is our purpose to establish some new characterizations of convex sets by means of local properties and to derive as a consequence certain known results. This will be done for sets in a topological linear space 7, such a space being a real linear space with a Hausdorff topology such that the operations of vector addition x+y and scalar multiplication ax are continuous in both variables jointly [3]. The principal results are contained in Theorems 4 and 5. In order to describe matters simply, the following notations are used. Notations. The interior, closure, boundary and convex hull of a set 5 in 7 are denoted by int S, cl .S, bd 5 and conv 5 respectively. The closed line segment joining xES and yES is indicated by xy, whereas L(x, y) stands for the line determined by x and y. The interior of a set 5 relative to the minimal linear variety containing it is denoted by intv 5. Set union, intersection and difference are denoted by U, • and ~ respectively. Vector addition and subtraction are denoted by + and — respectively. We let 0 and stand for the empty set and the origin of 7 respectively. In the statements of theorems and definitions the names of previous authors are indicated for historical purposes. Definition 1. Let SEL. A point xEbd S is called a point of mild convexity of S if x is not the midpoint of any segment uv with Op^uv ~xCint 5. It is desirable to compare this definition with those given by Tietze [5] and by Leja and Wilkosz [4]. See also Kaufman [2]. For a brief summary of earlier results see Bonnesen and Fenchel [l, p. 7]. Definition 2. 7e/ xEbd S, where SEL. The point x is a point of weak or strong convexity of S, or a point of weak or strong concavity of S, if there exists a neighborhood N(x) of x and a linear functional f with f(x) = c such that the following conditions hold : (a) (Tietze). The point x is a point of weak convexity of S if f(y) >c with yEN(x)~x implies y ES. (For strong convexity replace f(y) >c by f(y)^c.) (b) (Leja and Wilkosz). The point x is a point of strong concavity of S if f(y) ^cwith y£7V(x)~x implies yES. (For weak concavity replace f(y)úcbyf(y)<c.)