Polytope and vertex points

Question

Suppose we have a closed polytope $P$ in $\Re^{N}$. Is each point of $P$ a linear combination of the vertices of $P$ (that is its extreme points)? In other words: are the vertices of $P$ a base for the points of $P$?

Answer 1

Even more is true. Every $x\in P$ is a convex combination of the extreme points $x_e \in P$: \begin{align} \sum_e \lambda_e x_e &= x\\ \sum_e \lambda_e &= 1\\ \lambda_e &\ge 0 \end{align}