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We give a "convex" characterization to the following smoothness property, denoted by (C/): every compact convex set is the intersection of balls containing it. This characterization is used to give a transfer theorem for property (C/). As an application we prove that the family of spaces which have an equivalent norm with property (CI) is stable under Co and l p sums for 1 < p < oo. We also prove that if X has a transfinite Schauder basis, and Y has an equivalent norm with property (CI) then the space X® p Y has an equivalent norm with property (C/), for every tensor norm /?. Similar results are obtained for the usual Mazur property (/), that is, the family of spaces which have an equivalent norm with property (/) is stable under c 0 and l p sums for 1 < p < oo.