Closed cones and exposed faces I was wondering if a closed cone $C$ in a Banach space $X$ of dimension at least two always has an exposed face, that is, a face $F$ such that $F=C\cap\ker\phi$ for some positive $\phi\in X^*\setminus\{0\}$.
Thanks a lot for your help!
 A: I was trying to find some information on exposed faces and found this problem, so in the category better late then never: The answer is yes, in fact, every element in the boundary lies in an exposed face.
Let $x\in\partial C$. By the Hahn-Banach separation theorem there is a continuous linear functional $\varphi$ and a $t\in\mathbb{R}$ such that 
$$\varphi(x)\leq t<\varphi(y)\qquad y\in C^{\circ}.$$
Now let $y\in C^{\circ}$. Note that $\lim_{n\rightarrow\infty}\varphi(\frac{1}{n}y)=0$ so $t\leq0$. Furthermore suppose that $\varphi(y)<0$, then for $n$ large enough $\varphi(ny)<t\leq0$, which is a contradiction, so $\varphi$ is a positive linear functional.
From this it follows that $\emptyset\neq F=\text{ker}(\varphi)\cap C\subset\partial C$. Finally let $x,y\in C$ such that $\lambda x+(1-\lambda)y\in F$, then $\varphi(\lambda x+(1-\lambda)y)=0$ and as $\varphi(x),\varphi(y)\geq0$ we find that $\varphi(x)=\varphi(y)=0$, so $F$ is a face.
A: If $\phi \in X^*$ is positive, then $C \cap \ker(\phi)$ is automatically a face, which is furthermore exposed.¹ It is proper if and only if $\phi \neq 0$. So the question is: are there non-trivial positive functionals? This is answered by basic duality: since $C$ is closed and convex, it it also weakly closed (see e.g. [1, Thm 3.12], [2, IV.3.1], or [3, Thm 8.9(1)]). Then, under the dual pair $\langle X,X^*\rangle$, we know that $C$ is equal to its double dual wedge $C^{**}$ (by the one-sided bipolar theorem [2, Thm IV.1.5] or the wedge duality theorem [3, Thm 2.13(3)]). So $C^* = \{0\}$ if and only if $C = X$. We conclude: a closed wedge $C \subseteq X$ has a non-trivial exposed face if and only if $C \neq X$.
¹: Most authors include $C$ and $\varnothing$ among the exposed faces, so $C \cap \ker(0)$ is also an exposed face. 
References.
[1]: Walter Rudin, Functional Analysis, Second Edition (1991), McGraw–Hill.
[2]: H.H. Schaefer, with M.P. Wolff, Topological Vector Spaces, Second Edition (1999), Graduate Texts in Mathematics 3, Springer.
[3]: Charalambos D. Aliprantis, Rabee Tourky, Cones and Duality (2007), Graduate Studies in Mathematics 84, American Mathematical Society.
