Operations Preserving Convexity of Functions on $\mathbb{R}^n$: Affine Functions, Sums, Nonnegative Multiples, Pointwise Suprema and Affine Substitution

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$ be convex; convexity of functions is that of Convex Function on a Convex Subset of $\mathbb{R}^n$ (i).

(i) Affine functions. For every $a\in\mathbb{R}^{n}$ and $\beta\in\mathbb{R}$, the function $C\to\mathbb{R}$, $x\mapsto a\cdot x+\beta$, is convex.

(ii) Sums. If $f,g:C\to\mathbb{R}$ are convex, then so is $f+g:C\to\mathbb{R}$, $x\mapsto f(x)+g(x)$.

(iii) Nonnegative multiples. If $f:C\to\mathbb{R}$ is convex and $\lambda\in\mathbb{R}$ with $\lambda\ge0$, then $\lambda f:C\to\mathbb{R}$, $x\mapsto\lambda f(x)$, is convex.

(iv) Pointwise suprema. Let $\mathcal{F}$ be a nonempty set of convex functions $C\to\mathbb{R}$ such that, for every $x\in C$, the set $\{f(x):f\in\mathcal{F}\}$ is bounded above. Then the function $F:C\to\mathbb{R}$,

$$F(x)=\sup\{f(x):f\in\mathcal{F}\},$$

is convex, the supremum existing by The Real Numbers Form an Ordered Field in Which Every Nonempty Set Bounded Above Has a Supremum (ii).

(v) Affine substitution. Let $m\in\mathbb{N}$, $A\in\mathcal{M}_{n\times m}(\mathbb{R})$ and $b\in\mathbb{R}^{n}$, with the matrix-vector products $Ay$, let $f:C\to\mathbb{R}$ be convex, and let

$$C'=\{y\in\mathbb{R}^{m}:\ Ay+b\in C\},$$

which is convex by Operations Preserving Convexity in $\mathbb{R}^n$: Intersections, Translates, Scalar Multiples, Sums, Affine Images and Preimages, Closure and Interior (vii). Then the function $C'\to\mathbb{R}$, $y\mapsto f(Ay+b)$, is convex.