We give a “convex”
characterization to the following smoothness property, denoted by (CI): every
compact convex set is the intersection of balls containing it. This characterization is
used to give a transfer theorem for property (CI). As an application we prove that
the family of spaces which have an equivalent norm with property (CI) is stable
under c0 and lp sums for 1 ≤ p < ∞. We also prove that if X has a transfinite
Schauder basis, and Y has an equivalent norm with property (CI) then the space
X⊗pY has an equivalent norm with property (CI), for every tensor norm
ρ.
Similar results are obtained for the usual Mazur property (I), that is, the family
of spaces which have an equivalent norm with property (I) is stable under c0 and lp
sums for 1 < p < ∞.
