What is the geometrical meaning of the below theorem?
Let $K$ be a closed convex cone in euclidean space $V$, and let $b\in V$, $b\notin K$. Then there exists a nonzero vector $u\in V$ such that \begin{equation*} \langle{x,u}\rangle\geq 0 \text{ for every } x\in K, \text{ and } \langle{b, u}\rangle<0. \end{equation*}
The theorem says that given a closed convex cone $K$ in $\mathbb{R}^n$ and a point $b$ that's not in the cone, there exists a hyperplane (defined by its normal vector $u$) so that the cone is completely on one side of the hyperplane, and the point $b$ is on the other side. Here's a picture:
Note that the inner product $\langle{u,x}\rangle\geqslant0$ since the angle between $u$ and $x$ is less than $\pi$ radians. In contrast, $\langle{u,b}\rangle<0$, since the angle is greater than $\pi$ radians.
The books I have used for convex geometry (with applications to optimization) are Boyd's book Convex Optimization, and Nemhauser and Wolsey's book Integer and Combinatorial Optimization.