Problem: Let $x \in P, P$ a polyhedron. Show that if $x$ cannot be written as a convex combination of other points in $P$, then $x$ is an extreme point of $P$.
Proof: We see that in $x$, (at least) $n$ valid inequalities defining the polyhedron have to be tight (else, there exists some v such that $x ± λ$ v ∈ P for some small $λ$ and some vector $v$). Let these valid inequalities be $a_1x ≤ \beta_1,··· ,a_nx ≤ \beta_n$. So the the objective function $∑n_i=a_i$ ai has objective value $∑n_i=\beta _i$ at $x$ and is strictly smaller for all other points.
Hello, I am having an hard time getting an intuitive understanding of linear programming. In this proof, I do not understand why the fact that $x$ can not be written as convex combinations of other points in $P$, implies that we can find $n$ valid inequalities tight on $x$.