In linear programming, if we model the solution as a polytope, every vertex is a solution.
I'm trying to prove the following statement:
If $x$ and $y$ are two solutions with $n-1$ joint constraints, there exists a solution vector $c$ such that, for each $v\in P$ (the polytope of the problem) that is different from $x$ and from $y$, we have $\langle c,x\rangle = \langle c,y\rangle < \langle c,z\rangle$.
Basically, this means that only $x$ and $y$ are the optimal solutions for the problem. But why is it true? And what is the vector $c$ that exists here?