In the book of Papadimitriou. Combinatorial optimization, there is the instance of Linear Programming (LP) s.t.
\begin{equation}
\begin{split}
&{\rm minimize:\;} c_1x_1 + c_2x_2 + c_3x_3,\\
&{\rm subect\; to:\; }x_1 + x_2 + x_3 = 2
\end{split}
\end{equation}
The authors argue that the minimum is found at the corners of the 3d triangle (see figure) $v_1,v_2,v_3$.
I have a couple of questions:
If the $c_i$'s are all positive, then the minimum of the cost function is simply $2c_k$, where $0\leq c_k\leq c_j \leq c_i$, right?
If for instance two $c_i$'s are negative, then one has to pick the two $x_i$'s that multiply them, s.t. $x_i+x_j = 2$, right? In this case, it cannot be that only $v_1,$v_2$ or $v_3$ are the only solutions to the LP program.
