proper coneProperCone2004-09-20 13:58:062004-09-21 20:35:40Definition% this is the default PlanetMath preamble. as your knowledge
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% define commands hereA \emph{proper cone} is a \PMlinkname{cone}{Cone3} $C\subset\mathbb{R}^n$ that satisfies the following:
\begin{itemize}
\item $C$ is convex;
\item $C$ is closed;
\item $C$ is solid, meaning it has nonempty interior;
\item $C$ is pointed, meaning $x, -x\in C\Rightarrow x=0$.
\end{itemize}
\bigskip
A proper cone $C$ induces a partial ordering on $\mathbb{R}^n$:
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a\preceq b\Leftrightarrow b-a\in C.
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This ordering has many nice properties, such as transitivity, reflexivity, and antisymmetry.
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\begin{thebibliography}{4}
\bibitem{boyd} S. Boyd, L. Vandenberghe, \emph{Convex Optimization}, Cambridge University Press, 2004.
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