Let $C$ be a convex subset of $\mathbb{R}^n$. (and $0$ is supposed in $C$.)
I've read in the "Convex Optimization Theory" book of Bertsekas (page 25) that this set: $$X = \left\{\sum_{i=1}^m \alpha_i z_i ~ | ~\sum_{i=1}^m \alpha_i< 1, ~\alpha_i>0, ~ z_i \in C, ~ i=1,\dots,m\right\}$$
is included in $C$ since $C$ is convex. Firstly, I am not pretty sure what X is geometrically. And secondly, I can't find the proof of this assertion. Could someone help me, I would really appreciate.
The important part of this theorem is that $0\in C$. Let $x=\sum^{m}_{i=1}\alpha_{i}z_{i}\in X$ and let $\alpha_{0}=1-\sum^{m}_{i=1}\alpha_{i}$ and $z_{0}=0$. Then $$x=\sum^{m}_{i=0}\alpha_{i}z_{i}$$ and $\sum^{m}_{i=0}\alpha_{i}=1$. Geometrically it seems that $X$ is the interior of $C$ if $0\in C^{\circ}$ and it is the interior of $C$ plus the faces in which $0$ lies if $0$ lies on the boundary of $C$.