Convex Functions on $\mathbb{R}^n$: Standing Notation

(i) Base. This setting extends Separation in $\mathbb{R}^n$: Standing Notation: its conventions and notation are in force.

(ii) Points of $\mathbb{R}^{n+1}$ as pairs. For $x\in\mathbb{R}^{n}$ and $\mu\in\mathbb{R}$, $(x,\mu)$ denotes the point $\iota\bigl(x,(\mu)\bigr)$ of $\mathbb{R}^{n+1}$, where $(\mu)\in\mathbb{R}^{1}$ is the point whose single component is $\mu$ and $\iota:\mathbb{R}^{n}\times\mathbb{R}^{1}\to\mathbb{R}^{n+1}$ is the concatenation map, with $\mathbb{R}^{n}\times\mathbb{R}^{1}$ in place of $\mathbb{R}^{m}\times\mathbb{R}^{n}$ there; thus $(x,\mu)_{k}=x_{k}$ for $k\in[n]$ and $(x,\mu)_{n+1}=\mu$. Background: every point of $\mathbb{R}^{n+1}$ is $(x,\mu)$ for exactly one $x\in\mathbb{R}^{n}$ and one $\mu\in\mathbb{R}$, by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (iii), a point of $\mathbb{R}^{1}$ being determined by its single component (Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iii)); for $x,x'\in\mathbb{R}^{n}$ and $\mu,\mu',t\in\mathbb{R}$,

$$(x,\mu)+(x',\mu')=(x+x',\mu+\mu'),\qquad t(x,\mu)=(tx,t\mu),\qquad (x,\mu)-(x',\mu')=(x-x',\mu-\mu'),$$

by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (iv); and

$$(x,\mu)\cdot(x',\mu')=x\cdot x'+\mu\mu',\qquad \lVert(x,\mu)\rVert^{2}=\lVert x\rVert^{2}+\mu^{2},$$

by Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space (v), the dot product of $(\mu)$ and $(\mu')$ in $\mathbb{R}^{1}$ being the one-term sum $\mu\mu'$ (The Dot Product on $\mathbb{R}^n$ (i), Iterated Operations: Finite Sums and Finite Products (i), Iterating a Binary Operation along a Finite List (i)), so that $\lVert(\mu)\rVert^{2}=(\mu)\cdot(\mu)=\mu^{2}$ by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i).