Is $L_p(X, \mu, E)$ uniformly convex for $p \in (1, \infty)$ if $E$ is a uniformly convex Banach space?

Question

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite complete measure space, $(E, |\cdot|)$ a Banach space, and $p \in (1, \infty)$. Let $L_p := L_p(X, \mu, E)$ be the Bochner space of all $\mu$-integrable functions $f:X \to E$. Here we use Bochner integrals. If $E = \mathbb R$ then $L_p$ is uniformly convex for $p \in (1, \infty)$.

Is $L_p$ uniformly convex if $E$ is uniformly convex for $p \in (1, \infty)$?

Any reference for the proof is greatly appreciated.

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I saw this question on MSE which received one answer whose author does not remember the title of the reference paper. So I post it here.

Answer 1

A reference for this result would be Some more uniformly convex spaces by Mahlon M. Day, Bull. Amer. Math. Soc. 47(6): 504-507 (June 1941). (Alternative link at Project Euclid)