Let $E$ be a Banach space and $p \in (1, \infty)$. Is $L_{p}(X, \mu, E)$ uniformly convex?

Question

Let $(X, \Sigma, \mu)$ be a $\sigma$-finite measure space, $E$ a Banach space, and $f, g \in L_{p}(X, \mu, E)$. Here we use the Bochner integral. If $E = \mathbb R$, then

• $$ \left\|\frac{f+g}{2}\right\|_{L_{p}}^{p}+\left\|\frac{f-g}{2}\right\|_{L_{p}}^{p} \leq \frac{1}{2}\left(\|f\|_{L_{p}}^{p}+\|g\|_{L_{p}}^{p}\right) \quad \forall 2 \leq p<+\infty, $$
• $$ \left\|\frac{f+g}{2}\right\|_{L^{p}}^{q}+\left\|\frac{f-g}{2}\right\|_{L^{p}}^{q} \leq\left(\frac{1}{2}\|f\|_{L_{p}}^{p}+\frac{1}{2}\|g\|_{L_{p}}^{p}\right)^{q/p} \quad \forall 1<p<2. $$ where $$ \frac{1}{p}+\frac{1}{q}=1. $$

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Above are Clarkson's inequalities that help establish the uniform convexity of $L_{p}(X, \mu, \mathbb R)$ for $p \in (1, \infty)$.

For $p \in (1, \infty)$, I would like to ask if $L_{p}(X, \mu, E)$ is proved to be uniformly convex given a general Banach (or Hilbert) space $E$.

Answer 1

This is not true. If $E$ is $\mathbb R^2$ with the $\infty$-norm, $X=\{x\}$ is a one-point space, and $\mu$ is the counting measure on $X$, then $L_p(X,\mu,E)$ is isometrically isomorphic to $E$, which is not uniformly convex.