Separating and supporting hyperplane theorems I am having trouble understanding  the separating and supporting hyperplane theorems. I've read what I can online but am just not able to develop any intuition. Can someone please give a basic outline of these theorems using economic context? And any graphical interpretations would be extremely helpful.
I will sketch a solution for your question on the separating hyperplane theorem together with one economic application to producer theory (for more details please look at MWG). 

Theorem: Suppose $B \subset \mathbb{R}^N$ is a convex and closed and that $x \notin B$. Then there is a $p \in \mathbb{R}^N$ with $p \neq 0 $ and a value $c \in \mathbb{R}$ such that $px&gt;c$ and $py &lt; c$ for every $y \in B$

Application (Producer theory): Suppose the production set $Y$ is closed and convex. Then every efficient production $y \in Y$ is a profit maximizing production for some nonzero price vector $p \geq 0 $. 

Sketch of proof: 
Start with an efficient $y$ and define $P_y$ as the "better than" set with respect to $y$. This set is clearly convex (pick $y', y'' \in P_y$ and their convex combination is clearly "in it"). Next realize that because $y$ is efficient $P_y \cup Y = \emptyset$. We know have all the ingredients to apply our theorem above, and in particular $\exists p \neq 0$ such that $p y' \geq py'''$ $\forall$ $y' \in P_y$ and $\forall$ $y''' \in Y$. Because $y'$ can be chosen arbitrarily close to $y$ (remember $y' \in P_y$) we can straightforwardly show that $y$ is profit maximizing for $p$ 

Other "famous applications" that you may want to look at are:  

1) The "Second Fundamental theorem of Welfare Economics

2) The saddle point approach to dynamic programming developed by Karlin and Uzawa.