A topological characterization for closed sets under polar duality in \({\mathbb{Q}{}}^ n\) (Q1174458)

The \(\#\)-dual \(S^ \#\) of a subset \(S\) of \(\mathbb{Q}^ n\) consists of all \(y\in\mathbb{Q}^ n\) such that \(x^ Ty\geq 0\) for all \(x\in S\). A set \(S\) is called \(\#\)-constrained if \(S=T^ \#\) for some \(T\), and called \(\#\)- closed if \(S=S^{\#\#}\); it is easily seen that a set is \(\#\)-closed if and only if it is \(\#\)-constrained. The object of this note is to show that a set \(S\subseteq\mathbb{Q}^ n\) is \(\#\)-closed if and only if \(S=K\cap\mathbb{Q}^ n\) for some (topologically) closed cone \(K\) in \(\mathbb{R}^ n\) where the cone \(K\) has a linear basis consisting of rational vectors. A simple example shows that the latter condition on \(K\) cannot be omitted, and the authors note that \(K\) may always be taken as the topological closure of \(S\) in \(\mathbb{R}^ n\).