A NOTE ON CONVEXITY IN EUCLIDEAN n-SPACE

If a subset S of a real linear space is convex (with every two points Xl, X2 of S, the entire line segment Xx1+ (1 -X)x2, 0< X < 1, is contained in S) then it is also closed under the operation of taking finite convex linear combinations. For any set R in the space, the set of all finite convex linear combinations is the convex hull of R (the smallest convex set containing R). If the space has finite dimension n, it is well known that it is enough to mix at most n+1 points at a time to get the convex hull. If the space is topological, convergent mixtures of countably many points of R, as well as mean values of arbitrary measures on R, may also be taken. The question of how much more than the convex hull one gets thereby arises. It is well known that in an arbitrary locally convex topological space one stays in the closure of the convex hull. We show in this paper that in Euclidean n-space the interesting and useful fact is that one never gets out of the convex hull itself. A counter-example in Hilbert Space shows that infinite mixtures may take one outside the convex hull when the space is not finite dimensional. It will suffice to state our theorem in the following form.