Relative rotundity in \(L^ p(X)\) (Q1894901)

It is proved that, for \(1< p< \infty\), the Lebesgue-Bochner function space \(L^ p(X)\) is uniformly rotund relative to \(L^ p(Y)\) if and only if the normed space \(X\) is uniformly rotund relative to its linear subspace \(Y\). The proof, rather technical although clear, extends to Köthe normed spaces of vector-valued functions and to Day's substitution spaces.