Convex Combination of Finitely Many Points of $\mathbb{R}^n$

Let $n$ and $N$ be natural numbers with $1\le n$ and $1\le N$, let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and let $[N]$ and $[n]$ be the initial segments determined by $N$ and by $n$. Let $x:[N]\to\mathbb{R}^n$ be a family of points of Euclidean space, with values written $x_k$ and coordinates $(x_k)_j$ for $j\in[n]$, and let $t:[N]\to\mathbb{R}$ be a family with values written $t_k$. All sums below are the finite sums of the field $\mathbb{R}$.

The family $t$ is a system of convex weights of length $N$ if $0\le t_k$ for every $k\in[N]$ and

$$\sum_{k=1}^{N}t_k=1 .$$

For such a family $t$, the convex combination of $x_1,\dots,x_N$ with weights $t_1,\dots,t_N$, written $\sum_{k=1}^{N}t_kx_k$, is the point of $\mathbb{R}^n$ whose $j$th coordinate, for $j\in[n]$, is

$$\Bigl(\sum_{k=1}^{N}t_kx_k\Bigr)_j=\sum_{k=1}^{N}t_k\,(x_k)_j .$$