Let be a nonempty conver set in a vector space X, and let ro € 22. Assume furthermore that core(12) # 0. Then 2 and {xo} can be separated if and only if they can be properly separated. Proof. It suffices to prove that if N and {30} can be separated, then they can be properly separated. Choose a nonzero linear function f: X → R such that f(x) < f(xo) for all re. = Let us show that there exists w El such that f(w) < f(20). Suppose on the contrary that this is not the case. Then f(x) = f(xo) for all x E 12. Since core(52) = 0, by Lemma 2.47, the function f is the zero function. This contradiction completes the proof of the proposition.
