Analytic characterizations of Mazur's intersection property via convex functions (Q425745)

Let \(X\) be a Banach space. For a closed and bounded set \(B \subset X\), let \(\delta_B(x) = 0\) if \(x \in B\) and \(\delta_B(x) = \infty\) otherwise. This is a lower semi-continuous coercive proper convex function. For a closed ball \(B \subset X \times \mathbb R\), the ball projection function \(g\) is defined as \(g(x) = \inf\{r : (x,r) \in B \}\) when \(x\) is such that there exists an \(r\) with \((x,r) \in B\) and \(\infty\) otherwise. The authors prove the very interesting result that for a closed and bounded set \(B \subset X\), if there is a family \(G\) of ball projections below \(\delta_B\) such that \(\delta_B(x) = \sup_{g \in G} g(x)\), then \(B\) is an intersection of closed balls. Thus if this happens for all closed and bounded sets \(B \subset X\), then \(X\) has the Mazur intersection property (MIP). Conversely, if \(X\) has the MIP, the authors show that for any closed and bounded set \(B \subset X\) and for any lower semi-continuous proper convex function \(f\), there is a family \(G\) of ball projections below \(f\) such that \(f(x) + \delta_B(x) = \sup_{g \in G}g(x)\), for all \(x \in X\).NEWLINENEWLINEThe authors also characterize spaces \(X\) in which every compact convex set is an intersection of balls (CIP) in terms of the conjugate function of a ball projection in \(X\). For a lower semi-continuous convex function \(f\) on \(X^\ast\), with compact effective domain, if \(H_f\) denotes the set of ball conjugate functions above \(f\), then \(X\) has the CIP if and only if \(f^\ast = \inf\{h: h \in H_f\}\).