Suggest edit — convex set

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---
title: "convex set"
type: concept
related: [Convex Functions, Convex Sets, Perspective Function, Standard Definitions]
source: https://www.jemoka.com/posts/kbhconvex_sets/
confidence: high
status: active
---

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} + \qty(1-\theta) 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 \(x_1, x_2 \in C, 0 \leq \theta \leq 1 \implies \theta x_1 + \qty(1-\theta) x_2 \in C\) 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, \(f\qty(x) = Ax + b\) 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\qty(S) = \qty {f\qty(x) \mid x \in S} \text{ is cvx } \end{equation}
The inverse image of \(f^{-1}\qty( C)\) of a convex set \(f\) is convex:
\begin{equation} C \subseteq \mathbb{R}^{m} \text{ is cvx } \implies f^{-1}\qty( C) = \qty {x \in \mathbb{R}^{n} \mid f\qty(x) \in C} \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\qty(x) = \frac{Ax + b}{ c^{T}x + d} \end{equation}
where:
\begin{equation} \text{dom} f = \qty {x \mid c^{T} x + d &gt; 0} \end{equation}
linear-fractional function and its inverse image preserves convexity.