hyperplanes & halfspaces

hyperplanes & halfspaces

• a hyperplane in $R^n$ is a set of the form: $H=\left\{x\in {\mathbb{R}}^n:a^{\top }x=b\right\}$, $a\in R^n,a\ne 0,b\in R$
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• can be thought of as a hyperplanes as the level sets of linear functions
• when b=0, it is a set orthogonal to a
• hyperplane separate the space into 2 regions, which are called half space
  • $H_{-}=\{a^{T}x\le b\}$ → closed halfspace
  • $H_{++}=\{x:a^{T}x>b\}$ → open halfspace