Convex Weights, the Standard Simplex, and Convex Combinations in $\mathbb{R}^n$

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, let $N\in\mathbb{N}$.

(i) Convex weights and the standard simplex. The standard simplex is

$$\Delta_{N}=\Bigl\{t\in\mathbb{R}^{N}:\ t_{k}\ge0\text{ for every }k\in[N],\ \sum_{k=1}^{N}t_{k}=1\Bigr\},$$

and its points are called convex weights of length $N$.

(ii) Convex combination. For points $x_{1},\dots,x_{N}$ of $\mathbb{R}^{n}$, that is, a function $[N]\to\mathbb{R}^{n}$, $k\mapsto x_{k}$, and for $t\in\Delta_{N}$, the convex combination of $x_{1},\dots,x_{N}$ with weights $t$ is the point

$$\sum_{k=1}^{N}t_{k}x_{k}\in\mathbb{R}^{n},$$

the sum of the scalar multiples $t_{k}x_{k}$.

(iii) Convex combinations of points of a set. For $S\subseteq\mathbb{R}^{n}$, a point $z\in\mathbb{R}^{n}$ is a convex combination of $N$ points of $S$ if $z=\sum_{k=1}^{N}t_{k}x_{k}$ for some points $x_{1},\dots,x_{N}$ of $S$ and some $t\in\Delta_{N}$.