The Generalised (Uniform) Mazur Intersection Property

Given a family $\mathcal{C}$ of closed bounded convex sets in a Banach space $X$, we say that $X$ has the $\mathcal{C}$-MIP if every $C \in \mathcal{C}$ is the intersection of the closed balls containing it. In this paper, we introduce a stronger version of the $\mathcal{C}$-MIP and show that it is a more satisfactory generalisation of the MIP inasmuch as one can obtain complete analogues of various characterisations of the MIP. We also introduce uniform versions of the (strong) $\mathcal{C}$-MIP and characterise them analogously. Even in this case, the strong $\mathcal{C}$-UMIP appears to have richer characterisations than the $\mathcal{C}$-UMIP.