convex combination of corner point

Question

Assume that we have a linear programming problem. Can be written every point in feasible area as a convex combination of corner point?

Answer 1

If the feasible set is bounded/compact, then yes!

This is because for linear programming problems, the feasible set is always a convex set.

Then the Krein-Milman theorem states that the feasible set is the set of convex combinations of itsextreme points.

In this answer i assumed that what you called a corner point is an extreme point.

Answer 2

Depending on your definitions, the answer can well be no.

$S=\left\{x |\; x \geq 0 \right\} $

The only extreme point of this feasible set is $x=0$, and it is not possible to write all of the points in $S$ as convex combinations of the extreme points.

In general, you can write any point in the feasible set of an LP as a convex combination of its extreme points plus an affine combination of its extreme rays.