Convex Combinations: Small Cases, Splitting Off the Last Point, Dropping a Zero Weight, Joining Two Combinations, and Membership in Convex Sets

In the setting of Convex Sets in $\mathbb{R}^n$: Standing Notation, with the standard simplices $\Delta_{N}$ and convex combinations $\sum_{k=1}^{N}t_{k}x_{k}$.

(i) One point. A point $t\in\mathbb{R}^{1}$ lies in $\Delta_{1}$ if and only if $t_{1}=1$. For $x_{1}\in\mathbb{R}^{n}$ and $t\in\Delta_{1}$,

$$\sum_{k=1}^{1}t_{k}x_{k}=x_{1}.$$

(ii) Two points. For $x_{1},x_{2}\in\mathbb{R}^{n}$, a point $t\in\mathbb{R}^{2}$ lies in $\Delta_{2}$ if and only if $t=(1-s,s)$ for some $s\in[0,1]$, and then

$$\sum_{k=1}^{2}t_{k}x_{k}=(1-s)x_{1}+sx_{2}.$$

(iii) Splitting off the last point. Let $N\in\mathbb{N}$, $x_{1},\dots,x_{N+1}\in\mathbb{R}^{n}$, $t\in\Delta_{N+1}$ and $z=\sum_{k=1}^{N+1}t_{k}x_{k}$. Then $0\le t_{N+1}\le1$. If $t_{N+1}=1$, then $z=x_{N+1}$. If $t_{N+1}<1$, then $s=1-t_{N+1}>0$, the point $\tau=(t_{1}/s,\dots,t_{N}/s)$ lies in $\Delta_{N}$, and, with $y=\sum_{k=1}^{N}\tau_{k}x_{k}$,

$$z=(1-t_{N+1})\,y+t_{N+1}\,x_{N+1}.$$

(iv) Dropping a zero weight. Let $N\in\mathbb{N}$, $x_{1},\dots,x_{N+1}\in\mathbb{R}^{n}$, $t\in\Delta_{N+1}$ and $j\in[N+1]$ with $t_{j}=0$, and let $g_{j}:[N]\to[N+1]\setminus\{j\}$ be the bijection of Bijections between Intervals: the Gap Map and the Reversal (i). Then $(t_{g_{j}(1)},\dots,t_{g_{j}(N)})\in\Delta_{N}$ and

$$\sum_{k=1}^{N+1}t_{k}x_{k}=\sum_{k=1}^{N}t_{g_{j}(k)}x_{g_{j}(k)}.$$

(v) Joining two combinations. Let $N,P\in\mathbb{N}$, $x_{1},\dots,x_{N}\in\mathbb{R}^{n}$, $x'_{1},\dots,x'_{P}\in\mathbb{R}^{n}$, $\tau\in\Delta_{N}$, $\sigma\in\Delta_{P}$ and $\lambda\in[0,1]$. Let $w_{1},\dots,w_{N+P}$ be the points with $w_{k}=x_{k}$ for $k\in[N]$ and $w_{N+j}=x'_{j}$ for $j\in[P]$, and let $t\in\mathbb{R}^{N+P}$ have $t_{k}=(1-\lambda)\tau_{k}$ for $k\in[N]$ and $t_{N+j}=\lambda\sigma_{j}$ for $j\in[P]$. Then $t\in\Delta_{N+P}$ and

$$(1-\lambda)\sum_{k=1}^{N}\tau_{k}x_{k}+\lambda\sum_{j=1}^{P}\sigma_{j}x'_{j}=\sum_{k=1}^{N+P}t_{k}w_{k}.$$

(vi) Membership. Let $C\subseteq\mathbb{R}^{n}$ be convex, $N\in\mathbb{N}$, $x_{1},\dots,x_{N}\in C$ and $t\in\Delta_{N}$. Then $\sum_{k=1}^{N}t_{k}x_{k}\in C$.