Trying to make sense of convex combinations of more than 2 points I know that a line segment from point $a$ to point $b$ can be defined by $\{x:x=\alpha a + \beta b\}$ such that $\alpha + \beta = 1$. That is, the convex combination of $\{a, b\}$. We can generalize this to $(n \gt 2)$ dimensions by adding more coefficients. $\{x\in R^n:\sum_{i=1}^n x_i\alpha_i=x\}$ such that $\sum \alpha_i = 1$. When I saw the formula, it didn't look intuitive so I gave a thought and would like someone to correct me.
We'll build these convex combinations recursively starting with a line segment. To build a 2-simplex, we can add a point and take combinations of that point with every other point on the line segment. To build a 3-simplex, take combinations of a new point and all points on the 2-simplex (which we know how to build) and so on.
In symbols ($x_i\in R^n$): 
$K_1=\{x_1\alpha+x_2\beta:\alpha+\beta=1\}$
$K_2=\{x_3\alpha+y\beta:y\in K_1 \text{ and } \alpha+\beta=1\}$
$\dots$
$K_n=\{x_{n+1}\alpha+y\beta:y\in K_{n-1} \text{ and } \alpha+\beta=1\}$

A correct reformulation of your definition– we have the following recursive definition for the convex hull determined by $n+1$-points in $\mathbb{R}^m$:


*

*For $x_0 \in \mathbb{R}^m$, define $K[\{x_0\}] = \{x_0\}$

*Given $K[S]$ for some set $S\subset \Bbb{R}^m$, define 
$$K[S\cup \{x_0\}] = \{x_{0}\alpha+y\beta:y\in K[S] \text{ and } \alpha+\beta=1\}$$


That is, for the vectors $x_0,\dots,x_n \in \Bbb{R}^m$ we have
$$
K[\{x_0,\dots,x_n\}] = 
\{x_{n}\alpha+y\beta:y\in K[x_0,\dots,x_{n-1}] \text{ and } \alpha+\beta=1\}
$$
In $\Bbb{R}^m$, the convex hull determined by $m+1$ points is referred to as an $m$-dimensional simplex.
