Separation in $\mathbb{R}^n$: Standing Notation

(i) Base. This setting extends Convex Sets in $\mathbb{R}^n$: Standing Notation: its conventions and notation are in force. In particular $x\cdot y$, $\lVert x\rVert$ and $d_{E}$ are the dot product, the Euclidean norm and the Euclidean distance on $\mathbb{R}^{n}$ in force there, and closed and compact subsets of $\mathbb{R}^{n}$ refer to $d_{E}$.

(ii) Dot product and norm. Background: for $x,y,z\in\mathbb{R}^{n}$ and $t\in\mathbb{R}$, the dot product is symmetric and bilinear, and $x\cdot x=\lVert x\rVert^{2}\ge0$ with $\lVert x\rVert=0$ only for $x=0$, by The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (i), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (ii), The Dot Product on $\mathbb{R}^n$ Is Symmetric, Bilinear and Positive Definite (iii), Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (i) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (iii); $d_{E}(x,y)=\lVert x-y\rVert$ and $\lVert tx\rVert=|t|\,\lVert x\rVert$, by Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (ii) and Properties of the Euclidean Norm on $\mathbb{R}^n$: Square, Distance, Vanishing, Coordinate Bounds, Homogeneity, Triangle Inequalities and Expansion of a Square (v); and $0\cdot x=0$, by The Origin of ℝⁿ Has Dot Product Zero with Every Point (i).

(iii) Unit sphere and closed balls. $\mathbb{S}=\{a\in\mathbb{R}^{n}:\lVert a\rVert=1\}$ is the unit sphere of $\mathbb{R}^{n}$. Background: $\mathbb{S}$ and every closed ball $\bar{B}_{d_{E}}(c,r)$, for $c\in\mathbb{R}^{n}$ and real $r\ge0$, are compact, by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere (iii) and The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere (ii).