Let A and B be nonempty, convex subsets in a normed linear space X with int A and int AB . Then A and B

Let A and B be nonempty, convex subsets in a normed linear space X with int A ‰  ˆ… and int A‹‚B ˆ…. Then A and B can be properly separated.
Frequently stronger forms of separation are required. Two sets A and B are strictly separated by a hyperplane Hf. if A and B lie in opposite open halfspaces defined by Hf c, that is,
f (x) The sets A and B are strongly separated by the hyperplane Hf c. if there exists some number ε such that
f (x) or equivalently

sUD )s) sup f(x) < inf f(y