(Solution) - Let A and B be nonempty convex subsets in a -(2025 Original AI-Free Solution)

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Let A and B be nonempty, convex subsets in a normed linear space X with int A ? ? and int A?B ?. 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) < c < f (y) for every x ? A, y ? B
The sets A and B are strongly separated by the hyperplane Hf c. if there exists some number ? such that
f (x) < c - ? < c + ? < f (y) for every x ? A,y ? B
or equivalently