Geometric interpretation of duality and Slater's condition I am trying to study about optimization problems, Lagrange duality and related topics. I came across some presentation on the net, which claims to show the geometric interpretation of the duality and Slater's condition for a simple problem with only a single constraint : 
$$\begin{equation*}
\begin{aligned}
& \underset{x}{\text{minimize}}
& & f_0(x) \\
& \text{subject to}
& & f_1(x) \leq 0.
\end{aligned}
\end{equation*}$$
Here is the following slide:

Now, I understand how $p^*$ is depicted: Primal problem has a constraint $f_1(x) \leq 0$ and we only consider negative $u$ values therefore. The point $(u,t)$ with the minimum $t$ value is picked where $u \leq 0$.
But I completely don't understand how I should interpret the dual function $g(\lambda)$ to begin with. $g(\lambda)$ is depicted as a line (hyperplane). But according to the definition of $g(\lambda)$ it must be a scalar value. The dual problem is $g(\lambda) = \inf_x(f_0(x) + \lambda f_1(x))$ where $\lambda \geq 0$. So, for a given $\lambda \geq 0$ we must go and seek a $(u,t)$ point in $G$ which minimizes $g(\lambda)$. How is this connected with a hyperplane to begin with? We are in $(u,t)$ space, which has no $\lambda$ parametrization in it. I direly need some clarifications here.

See lecture 8 of Stanford course convex optimization. At time 1.04 (one hour and 4 minutes from start) Stephen Boyd start explaining the geometric interpretation.
http://www.youtube.com/watch?v=FJVmflArCXc
You also have his book (free from his web site) with more formal details.
