TITLE: On the Proof of Fundamental Theorem of Linear Programming.
QUESTION [5 upvotes]: Having read the link:
Why maximum/minimum of linear programming occurs at a vertex?
I understand why the optimal solution of any linear programming problem must be on the corner or lies on a face of a convex polygon. But my question is about a proof given below:
http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_programming
I do not know why the next argument was
$$x^\ast - \frac{\epsilon}{2} \frac{c}{||c||} \in P$$
Sorry to ask this question, but I can't imagine where the term 
$$\frac{\epsilon}{2} \frac{c}{||c||}$$
is coming from. Anyone, please enlighten me. Thanks.
Isn't it the contradiction will still be true even if we have, 
$$x^\ast - \frac{c}{||c||}$$.
The problem may be this construction may not be in $P$. Now, I really want to visualize the expressions 
$\frac{\epsilon}{2} \frac{c}{||c||} $ and $\frac{c}{||c||}$. Thanks.

REPLY [3 votes]: In the proof, the assumption that $x^\ast$ is in the interior of $P$ is shown to lead to a contradiction.
Let 
$$z = x^\ast - \frac{\epsilon}{2} \frac{c}{||c||}$$
Then
$$||z - x^\ast|| = \frac{\epsilon}{2} \frac{||c||}{||c||}= \frac{\epsilon}{2} < \epsilon,$$
and
$$z \in B_{\epsilon}(x^\ast) \subset P.$$
This was chosen to obtain a feasible point $z \in P$, such that $c^Tz < c^Tx^\ast$
Since $x^\ast$ is optimal we have a contradiction, implying that $x^\ast$ cannot lie in the interior of $P$.