Operations Preserving Convexity in $\mathbb{R}^n$: Intersections, Translates, Scalar Multiples, Sums, Affine Images and Preimages, Closure and Interior

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

(i) Trivial cases. $\emptyset$ and $\mathbb{R}^{n}$ are convex.

(ii) Intersections. For every set $\mathcal{C}$ of convex subsets of $\mathbb{R}^{n}$, the intersection taken inside $\mathbb{R}^{n}$,

$$\bigcap\mathcal{C}=\{x\in\mathbb{R}^{n}:x\in K\text{ for every }K\in\mathcal{C}\},$$

is convex.

(iii) Translates. For every convex $C\subseteq\mathbb{R}^{n}$ and every $a\in\mathbb{R}^{n}$, the set $a+C=\{a+x:x\in C\}$ is convex.

(iv) Scalar multiples. For every convex $C\subseteq\mathbb{R}^{n}$ and every $\lambda\in\mathbb{R}$, the set $\lambda C=\{\lambda x:x\in C\}$ of the scalar multiples of points of $C$ is convex.

(v) Sums and differences. For all convex $C,D\subseteq\mathbb{R}^{n}$, the sets

$$C+D=\{x+y:x\in C,\ y\in D\}\qquad\text{and}\qquad C-D=\{x-y:x\in C,\ y\in D\}$$

are convex.

(vi) Affine images. Let $C\subseteq\mathbb{R}^{n}$ be convex, $m\in\mathbb{N}$, $A\in\mathcal{M}_{m\times n}(\mathbb{R})$ and $b\in\mathbb{R}^{m}$. With the matrix-vector products $Ax$, the set $\{Ax+b:x\in C\}\subseteq\mathbb{R}^{m}$ is convex.

(vii) Affine preimages. Let $m\in\mathbb{N}$, $A\in\mathcal{M}_{m\times n}(\mathbb{R})$, $b\in\mathbb{R}^{m}$ and let $E\subseteq\mathbb{R}^{m}$ be convex. With the matrix-vector products $Ax$, the set $\{x\in\mathbb{R}^{n}:Ax+b\in E\}$ is convex.

(viii) Closure. For every convex $C\subseteq\mathbb{R}^{n}$, the closure $\operatorname{cl}C$ is convex.

(ix) Interior. For every convex $C\subseteq\mathbb{R}^{n}$, the interior $\operatorname{int}C$ is convex.