Rotund Renormings in Spaces of Bochner Integrable Functions

We show that if μ is a probability measure and X is a Banach space, then the Lebesgue-Bochner space $L^1(\mu,X)$ admits an equivalent norm which is rotund (uniformly rotund in every direction, locally uniformly rotund, or midpoint locally uniformly rotund) if X does. We also prove that if X admits a uniformly rotund norm, then the space $L^1(\mu,X)$ has an equivalent norm whose restriction to every reflexive subspace is uniformly rotund. This is done via the Luxemburg norm associated to a suitable Orlicz function.