convex set — Jemoka Knowledge Base

constituents set C \theta \in \mathbb{R} requirements A set C is a convex set if:

\begin{equation} x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_{1} + \left(1-\theta\right) x_{2} \in C \end{equation}

definitions standard definitions additional information operations that preserve convexity Convex sets is a calculus! Methods to showing complexity: Anything in Euclidian Geometry Crash Course apply definition: show x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_1 + \left(1-\theta\right) x_2 \in C use convex functions show that C is obtained from convex sets via the following operations intersection Intersections of any number of convex sets, include infinite, are convex. affine mapping Suppose f : \mathbb{R}^{n} \to \mathbb{R}^{m} is affine, that is, f\left(x\right) = Ax + b for A \in \mathbb{R}^{m \times n} and b \in \mathbb{R}^{m}. The image of a convex set under affine f is convex:

\begin{equation} S \subseteq \mathbb{R}^{n} \text{ is cvx } \implies f\left(S\right) = \left\{f\left(x\right) \mid x \in S\right\} \text{ is cvx } \end{equation}

The inverse image of f^{-1}\left( C\right) of a convex set f is convex:

\begin{equation} C \subseteq \mathbb{R}^{m} \text{ is cvx } \implies f^{-1}\left( C\right) = \left\{x \in \mathbb{R}^{n} \mid f\left(x\right) \in C\right\} \text{ is cvx} \end{equation}

perspective mappings perspective function and its inverse image preserves convexity. linear-fractional function for f: \mathbb{R}^{n} \to \mathbb{R}^{m}:

\begin{equation} f\left(x\right) = \frac{Ax + b}{ c^{T}x + d} \end{equation}

where:

\begin{equation} \text{dom} f = \left\{x \mid c^{T} x + d > 0\right\} \end{equation}

linear-fractional function and its inverse image preserves convexity.