A convex set is a set of elements from a vector space such that all the points on the straight line line between any two points of the set are also contained in the set. If a and b are points in a vector space the points on the straight line between a and b are given by
The above definition can be restated as: A set S is convex if for any two points a and b belonging to S there are no points on the line between a and b that are not members of S. Another restatement of the definition is: A set S is convex if there are no points a and b in S such that there is a point on the line between a and b that does not belong to S. The point of this restatement is to include the empty set within the definition of convexity. The definition also includes singleton sets where a and b have to be the same point and thus the line between a and b is the same point.
With the inclusion of the empty set as a convex set then it is true that:
The proof of this theorem is by contradiction. Suppose for convex sets S and T there are elements a and b such that a and b both belong to S∩T, i.e., a belongs to S and T and b belongs to S and T and there is a point c on the straight line between a and b that does not belong to S∩T. This would mean that c does not belong to one of the sets S or T or both. For whichever set c does not belong to this is a contradiction of that set's convexity, contrary to assumption. Thus no such c and a and b can exist and hence S∩T is convex.
The significance of convex sets in economics is some theorems on the existence of separating planes and support planes for any convex set. The nature of these planes, more properly hyperplanes, will be explained later.
On a vector space there are linear functionals which map the vector space into the real numbers; i.e., f: V->R such that f(x+y)=f(x)+f(y). These linear functionals form a vector space, called the dual space to the original vector space. A hyperplane is the set points of the vector space that map into the same real value; i.e., x such that f(x)=b. The hyperplane has associated with it two open half spaces; i.e., the set of points such that f(x)<b and the set of points such that f(x)>b. There are also to closed half spaces associated with a hyperplane; i.e., the set of points such that f(x)≤b and the set of points such that f(x)≥b.
Let C be a nonempty convex set of a vector space V and y any point of V that does not belong to C. There exists a hyperplane g(x)=b such that y is in the hyperplane and C is a subset of one of the two open half spaces associated with the hyperplane; i.e., for all x belonging to C either g(x)<b or for all x belonging to C g(x)>b.
If y is a boundary point of a closed, nonempty convex set C then there exists a supporting hyperplane h(x)=b such that y is in the hyperplane, h(y)=b, and all of C lies entirely in one of the two closed half spaces associated with the hyperplane; i.e., either for all x in C, h(x)≤b or for all x in C, h(x)≥b.