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@jemoka / Jemoka Knowledge Base / wiki/concepts/seperating_hyperplane_theorem.md
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--- 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 \(a^{T}x \geq b, a^{T}x \leq b\).