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

(i) Base. This setting extends Multivariable Calculus: Notation and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: their conventions and notation are in force.

(ii) Segments and convex sets. For $x,y\in\mathbb{R}^{n}$, $[x,y]$ is the segment from $x$ to $y$, and a subset of $\mathbb{R}^{n}$ is convex as in Segments and Convex Subsets of $\mathbb{R}^n$ (ii).

(iii) Finite sums of points. For $N\in\mathbb{N}$ and a function $y:[N]\to\mathbb{R}^{n}$, $k\mapsto y_{k}$, the point $\sum_{k=1}^{N}y_{k}$ of $\mathbb{R}^{n}$ is the finite sum of $y$ for the sum of points of Euclidean Space $\mathbb{R}^n$: n-Tuples of Real Numbers with Their Componentwise Operations (iv), formed in the commutative ring $\mathbb{R}^{n}=\mathbb{R}^{[n]}$ of Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (i). Background: such sums are computed coordinatewise,

$$\Bigl(\sum_{k=1}^{N}y_{k}\Bigr)_{j}=\sum_{k=1}^{N}(y_{k})_{j}\qquad(j\in[n]),$$

by Rules for Pointwise Operations on Real-Valued Maps: a Commutative Ring, Scalar Multiples and Evaluation (vi).

(iv) Closure and interior. For $A\subseteq\mathbb{R}^{n}$, $\operatorname{cl}A$ and $\operatorname{int}A$ are the closure and the interior of $A$ in $(\mathbb{R}^{n},d_{E})$.

(v) Compactness. Background: for $q\in\mathbb{N}$, a subset of $\mathbb{R}^{q}$ is compact if and only if it is closed and bounded, by The Heine-Borel Theorem in $\mathbb{R}^n$, and Compactness of Closed Balls and the Unit Sphere (i).