Prove that $\{x: Ax<b\}$ has no extreme points

Question

Suppose we have the set {$x : Ax < b$} and we want to show that it has no extreme points. Graphically, and intuitively this would make sense to me because the points of intersection of all the linear equations aren't actually in the set. But I am having trouble showing this mathematically

Answer 1

I will assume that $A$ is a continuous linear map. We can write $x$ as $\frac {(1+\frac 1 n)x} 2+\frac {(1-\frac 1 n)} 2 x$ and the points ${(1+\frac 1 n)x} $, $ {(1+\frac 1 n)x} $ both belong to the given set if $n$ is sufficiently large. Hence $x$ is not an extreme point.