\(\DeclarePairedDelimiterX{\Set}[2]{\{}{\}}{#1 \nonscript\;\delimsize\vert\nonscript\; #2}\) \( \DeclarePairedDelimiter{\set}{\{}{\}}\) \( \DeclarePairedDelimiter{\parens}{\left(}{\right)}\) \(\DeclarePairedDelimiterX{\innerproduct}[1]{\langle}{\rangle}{#1}\) \(\newcommand{\ip}[1]{\innerproduct{#1}}\) \(\newcommand{\bmat}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\barray}[1]{\left[\hspace{2.0pt}\begin{matrix}#1\end{matrix}\hspace{2.0pt}\right]}\) \(\newcommand{\mat}[1]{\begin{matrix}#1\end{matrix}}\) \(\newcommand{\pmat}[1]{\begin{pmatrix}#1\end{pmatrix}}\) \(\newcommand{\mathword}[1]{\mathop{\textup{#1}}}\)
Real Inner Product
Vectors as Matrices
Real Hyperplanes
Needed by:
Real Convex Sets
Real Convex Sets and Halfspaces
Real Polyhedra
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Graph PDF

Real Halfspaces


For any $a \in \R ^n$ and $\alpha \in \R $, the sets

\[ \Set{x \in \R ^n}{a^\tp x \leq \alpha }, \quad \Set{x \in \R ^n}{a^\tp x \geq \alpha } \]

are called closed halfspaces and the sets

\[ \Set{x \in \R ^n}{a^\tp x < \alpha }, \quad \Set{x \in \R ^n}{a^\tp x > \alpha } \]

are called open halfspaces.

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