Q:

about an interesting affirmation involving convex sets

Consider the following definition
Definition: Let $\Omega \subset R^n$  a bounded convex set. A point $x \in \partial \Omega$ is called an extremal point if $x$ cannot be written as linear combination of the form $x = tx_1 + (1-t)x_2$ ,$t \in (0,1)$ , $x_1,x_2 \in \partial \Omega$.  The set formed by this elements will be denoted by $E_{\Omega}$.
I am reading an article, and  the author writed  this:
Let $\Omega \subset R^n$  a bounded convex set.If the point $x \in\partial \Omega$ is not an extremal point then $x$ can be written as a linear combination as above
$$ x = \displaystyle\sum_{i=1}^{n}t_i x_i \ \ where  \ \  \displaystyle\sum_{i=1}^{n}t_i =1, x_i \in \overline{E_{\Omega}}$$
I have no idea to prove this . I searched how but i dont find anything. someone can give me a reference or a proof, please? I am  an ignorant on theorems involving convex sets =\
Thanks in advance

A:

$$ \partial\Omega \subseteq \overline{\Omega} = \overline{\text{conv}(E_\Omega)} = \text{conv}(\overline{E_\Omega}) $$
The first equality is the finite-dimensional case (due to Minkowski) of the Krein-Milman theorem, that a compact convex set is the closed convex hull of its extreme points (together, I suppose, with the theorem that the closure of a convex set is convex).  The second equality, that closure commutes with convex hull, holds for bounded sets in $\mathbb{R}^n$ and is a standard application of Carathéodory's theorem.
Since you asked for references:

In Schneider's Convex Bodies: The Brunn-Minkowski Theory, the first equality is Corollary 1.4.5, the fact that the closure of a convex set is convex is part of Theorem 1.1.9 (whose proof Schneider omits as an "easy exercise"), and the second equality is Theorem 1.1.10.
In Lay's Convex Sets and their Applications, the first equality is Theorem 5.6, the fact that the closure of a convex set is convex is Theorem 2.11, and the second equality is... actually, I don't see it, but the hard direction ($\subseteq$) follows from Theorem 2.30.

