Gauge functions for convex cones (Q2922479)

Let \(X\) be a real normed space with dual \(X^*\), \(K\) a closed convex pointed (i.e., \(K\cap(-K)=\{0\}\)) cone with nonempty interior and \(K^+=\{x^*\in X^* : x^*(x)\geq 0\;\, \forall x\in K\}\) its dual cone. A gauge function for \(K\) is a functional \(\varphi\) of the form \(\varphi(x)=\sup\{x^*(x) : x^*\in C\}\) for some \(w^*\)-compact subset \(C\) of \(X^*\) such that \(0\notin C\) and the closed convex cone generated by \(C\) is \(K^+\). The author gives a characterization of gauge functions for \(K\) as those functions \(\varphi: X\to\mathbb{R} \) that satisfy the conditions (i) \(\varphi\) is a continuous sublinear functional; (ii) \(\varphi<0\) on the interior of \(-K\); (iii) \(\varphi>0\) on the complement of \(-K\). In this case, the set \(C\) is given by \(C=\{x^* \in X^* : x^*(x)\leq \varphi(x)\;\, \forall x\in X\}\).NEWLINENEWLINEAs an application, the author considers the oriented distance of \textit{J. B. Hiriart-Urruty} [Math. Oper. Res. 4, 79--97 (1979; Zbl 0409.90086)], \(\varphi(x)=d(x,K)-d(x,X\setminus(-K)),\) and proves the equality \(\varphi(x)=\sup\{x^*(x) : x\in C\}\), where \(C=\{x^*\in K^+ : \|x^*\|=1\}\).