We have a Convex hull of a set $X\subseteq R^{n}$ defined as $C$, we need to prove that $C$ can we written as the following: $$\bar{C}=\sum_{i=1}^mt_ix_i$$
where $m\geq 1,t_i\geq0, x_1,x_2,….,x_m\in X$ and $\sum_{i=1}^mt_i=1$.
So, we need to prove that $C=\bar{C}$.
We do that by proving $\bar{C}\subseteq C$ and $C\subseteq \bar{C}$.
I understood the first part, but got stuck at the second one.
So, in order to prove $C\subseteq \bar{C}$, the proof says, we need to prove that $\bar{C}$ is convex.
That actually makes sense, since we know $C$ is the convex hull and a convex hull is the intersection of all convex sets that contain $X$. That is, $C$ is contained in all convex sets that contain $X$.
But I don't see how $\bar{C}$ could contain $X$. If we take $m=1$, it could only contain a subset $\{x_1,x_2,…..,x_m\}$ of $X$ and not the whole set.
Am I missing something ? Kindly help !
Best Answer
$\overset {-} C$ contains all points of the form $\sum_{i=1}^{m} t_i x_i$ with $m \geq 1, t_i \geq o, x_i \in X$. All quantities here are variables, nothing is fixed. It is obvious that every $x \in X$ can be written in above form by taking $m=1,x_1=x,t_1=1$ so $\overset {-} C$ contains every point of $X$. ($x_1,x_2,..,x_m$ are not fixed elements of $X$).