Assume a combination of the form: $\vec{S}=\sum_ip_i\vec{v}_i$ with $0\le p_i\le 1$, what is this combination called?
And considering the set: $$\mathcal{S}=\{\sum_ip_i\vec{v}_i \mid 0\le p_i\le 1\}$$ What are its properties? For example, it's obviously convex, and for finite $\{\vec{v}_i\}$, it is a convex polytope. But how many vertices does it have? assuming there are $n$ different vectors $\{\vec{v}_i\}$