B Supporting Hyperplane of a Convex Set

A hyperplane is a generalization of a plane in a coordinate system of a number of dimensions. Any point y in a hyperplane satisfies a set of linear equations 

We will now show that if 5 is a convex set, then at each of its boundary point we have a supporting hyperplane, or equivalently p (y — y) 0 for aU y in S. In the following two steps we show that  

They are non-negative since they depend on non-negative AtiS, AtfS, Atf, Atf, and a. 

In a closed convex set 5 in I , there exists a unique point y, which is closest to a given point x outside the set S. 

Given any boundary point y of the set S, there exists a non-zero vector p such that the dot product p (y — y) 0 for all y in the S. This result means that a hyperplane supports the set S at each of its boundary point. 

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