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I need a little help in the following question.

Let $V$ be a real normed vector space and let $K\subset V$ be a compact and convex subset. Let $C\subset V^\prime$ be a convex cone. Assume that, for each $f\in C$, there exists $v_f\in K$ such that $f(v_f)\geqslant0$. Show that, there exists $v\in K$ such that $f(v)\geqslant0$, for all $f\in C$.

Hint: For $f\in C$, show that the collection of sets $K_f=\{v\in K;\ f(v)\geqslant0\}$ (which are closed and non-empty) has the finite intersection property.

Remark. We say that $C$ is a cone in a vector space if for all $v\in C$ and all $\lambda\geqslant0$, we have that $\lambda v\in C$.

My idea: Since $K$ is compact, if we show that the above collection has the finite intersection property, we get that the arbitrary intersection $$\bigcap_{f\in C}K_f$$ is non-empty, by using Theorem 1 of this page. Then it follows the result.

So, how we prove that $\{K_f\}_{f\in C}$ has the F.I.P.?

Mathecm
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1 Answers1

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Since you only need a little help, I will write a partial answer and let you figure out the rest:

Suppose that $K$ is a compact in $V={\mathbb R}^n$, $C$ is a closed convex cone in $V^*$. I will be using the standard identification of $V^{**}$ with $V$.

Let $\hat{K}$ denote the cone over $K$, this is again a closed convex cone. If there is no $v\in K$ such that $f(v)\ge 0$ for all $f\in C$, then $\hat{K}\cap C^*=\{0\}$, where $C^*\subset V^{**}=V$ is the dual cone of $C$. Thus (by a theorem which you should know from your class or the textbook you are reading), there exists a closed half-space $H\subset V$ containing $C^*$ and intersecting $\hat{K}$ only at the origin. Thus, since $C^{**}=C$, there exists $f\in C$ such that $$ H= \{x: f(x)\ge 0\}. $$ Therefore, $f|_{K -\{0\}}$ is strictly positive, implying that $0\in K$. A contradiction, since $c(0)=0\ge 0$ for all $c\in C$.

Moishe Kohan
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