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Given a cone \[K\]
\[y \in K = \{Ax|x\geq 0\}\]
The duality set \[K^{\ast}\] is defined as
\[x^{\ast} \in K^{\ast}=\{y^{T}x^{\ast}\geq 0\] for all \[y \in K \}\]
then
\[y^{T}x^{\ast}=(Ax)^{T}x^{\ast}=x^{T}(A^{T}x^{\ast})\geq 0\]
Since \[x \geq 0\], the duality set is
\[K^{\ast}=\{A^{T}x^{\ast}\geq0\}\]
which is a polyhedral cone (intersections of finite number of halfspaces that have corresponding halfplanes passing through origin.)
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