Abstract
A Banach space X has the asymptotic-norming property if and only if the Lebesgue-Bochner function space Lp (μ, X) has the asumptotic-norming property for p with 1 < p < ∞. It follows that a Banach space X is Hahn-Banach smooth if and only if Lp (μ, X) is Hahn-Banach smooth for p with 1 < p < ∞. We also show that for p with 1 < p < ∞, (1) if X has the compact Mazur intersection property then so does Lp(μ, X); (2) if the measure μ is not purely atomic, then the space Lp(μ, X) has the Mazur intersection property if and only if X is an Asplund space and has the Mazur intersection property.
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