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I'm trying to solve the following problem in Functional Analysis but I'm not sure how to do it. The only hint I got is that I will have to use the Minkowski Functional and Hahn-Banach Theorem to get this statement proved. But how? Can someone explain it to me or give me some hints?

Exercise: Let $C \neq \varnothing$ be a convex, not necessarily open subset of a $\mathbb R$-vector space $X$ so that each point $x \in C$ has the property that for any $y \in X \setminus\{x\}$ there exists $\epsilon > 0$ with $x+ty \in C$ for all $t \in \mathbb R$ with $|t| < \epsilon$. Show that for each point $y \in X \setminus C$ there exists a linear functional $f: X \to \mathbb R$ and $ \alpha \in \mathbb R$ with $f(x) < \alpha$ for any $x \in C$ and $\alpha = f(y)$.

user372904
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  • The Idea was the following: W.l.o.g. let $0 \in C$, $L=span(y)$ and define $f: L \to \mathbb R$ where $\lambda y \to \lambda p_c(y)$, define $p_c(x)=inf{a>0 s.t. a^{-1}*x \in C}$ . And then to show $C= {p_c = 1}$. And by using Hahn-Banach to get the statement. But how to show C is equal to the set as it is defined? And which role plays the Hahn-Banach here? – user372904 Oct 12 '16 at 08:35

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