Definition 1

Let $A$ be a convex set in an affine space $E$. Then we say that a point $x \in A$ is an extremal point of $A$ if there does not exist an open segment that is contained in $A$ and contains $x$.

In other words, the relations $x = \lambda y + (1 - \lambda)z,\ y \in A,\ z \in A,\ y \neq z$ and $0 \leq \lambda \leq 1$ imply $\lambda = 0$ or $\lambda = 1$ (thus $x = y$ or $x = z$). This implies that $x$ cannot be the barycentre of a set of $n$ points $x_i$ of $A$ carrying positive masses unless $x$ is one of the $x_i$; for this is just the definition when $n = 2$; for arbitrary $n$ argue by induction on $n$, as $x$ is the barycentre of $x_1$ and of the barycentre $y_1$ of the $x_i$ with $2 \leq i \leq n$, therefore $x$ is identical with $x_1$ or $y_1$, and in the second case it is sufficient to apply the induction hypothesis.

To say that $x$ is an extremal point of $A$ also means that $A - \{x\}$ is convex.