---
title: "seperating hyperplane theorem"
type: concept
related: [Convex Sets]
source: https://www.jemoka.com/posts/kbhseperating_hyperplane_theorem/
confidence: high
status: active
---

IF \(C\) and \(D\) are non-empty disjoint \(C \cap D = \emptyset\), and \(C\) and \(D\) are convex sets, \(\exists a \neq 0\) such that:
\begin{equation} a^{T} x \leq b, x \in C \end{equation}
\begin{equation} a^{T}x \geq b, x \in D \end{equation}
The hyperplane:
\begin{equation} \qty {x \mid a^{T} x = b} \end{equation}
separates \(C, D\) such that \(a^{T}x \geq b, a^{T}x \leq b\).
