Small Cases, Reduction, and Membership for Convex Combinations

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 for a natural number $p$ let $[p]$ be the initial segment determined by $p$. Regard Euclidean space $\mathbb{R}^n$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ of points and the scalar multiple $\lambda z$. Systems of convex weights and convex combinations are those of Convex Combination of Finitely Many Points of $\mathbb{R}^n$, and sums of real numbers are the finite sums of the field $\mathbb{R}$.

Then the following hold.

1. (Small cases) Let $x:[1]\to\mathbb{R}^n$ and let $t$ be a system of convex weights of length $1$. Then $t_1=1$ and $\sum_{k=1}^{1}t_kx_k=x_1$. Let $y:[2]\to\mathbb{R}^n$ and let $u$ be a system of convex weights of length $2$. Then $u_2=1-u_1$ and

$$\sum_{k=1}^{2}u_ky_k=u_1\,y_1+(1-u_1)\,y_2 .$$

2. (Reduction) Let $x:[N+1]\to\mathbb{R}^n$, let $t$ be a system of convex weights of length $N+1$, write $t'$ and $x'$ for the restrictions of $t$ and $x$ to $[N]$, and set $s=\sum_{k=1}^{N}t'_k$. Then $s=1-t_{N+1}$, $0\le s$ and $s\le1$. Moreover:

(a) if $t_{N+1}=1$, then $\sum_{k=1}^{N+1}t_kx_k=x_{N+1}$;

(b) if $t_{N+1}\ne1$, then $0<s$, the family $\tau$ on $[N]$ given by $\tau_k=s^{-1}t'_k$ is a system of convex weights of length $N$, it satisfies $s\,\tau_k=t'_k$ for every $k\in[N]$, and, with $y=\sum_{k=1}^{N}\tau_kx'_k$,

$$\sum_{k=1}^{N+1}t_kx_k=s\,y+(1-s)\,x_{N+1}.$$

3. (Membership) Let $C\subseteq\mathbb{R}^n$ be convex, let $x:[N]\to\mathbb{R}^n$ take all of its values in $C$, and let $t$ be a system of convex weights of length $N$. Then $\sum_{k=1}^{N}t_kx_k\in C$.