So the question is framed like this,
Let $C \subseteq \mathbb{R}^{n}$ be a convex set and D = {Ax+b | $x \in \mathbb{R}^{m}, A \in \mathbb{R}^{n\times m}, b \in \mathbb{R}^{n}$}. Also $C \cap D = \Phi$.
Show that there exist $\lambda \neq 0$ such that $A^{T}\lambda = 0$ and $\lambda^{T}y \leq \lambda^{T}b$, $\forall y \in C$.
$\textit{Solution}$
So set D is known to be convex and by separating hyperplanes theorem between two convex sets C1 and C2, which are disjoint , there exists a vector $a\neq0$ such that
$a^{T}x_{1} \leq a^{T}x_{2}$, $ \forall$ $x_{1} \in C1, x_{2} \in C2$.
$\therefore$ $\lambda^{T}y \leq \lambda^{T}(Ax+b)$
I cant seem to proceed further, how to solve it from this point?
There is also an extension to this problem which is as, Show that {x | Ax < b, $A \in \mathbb{R}^{n\times m}, x \in \mathbb{R}^{n}$} is empty if and only if {$\lambda \in \mathbb{R}^{m} | \lambda \neq 0, \lambda > 0, A^{T}\lambda = 0,\lambda^{T}b\leq0$} is not empty. We have show this by using the result of first part.
Thanks
