$L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$.

Question

How can I prove that $L_{p}(\mathbb{T})$ is not uniformly convex if $p \in \{1,\infty\}$. Here $\mathbb{T} = \mathbb{R}/\mathbb{Z}$

Answer 1

For example, you can take $f = \chi_{[0,2/3)}$ and $g = \chi_{[1/3,1)}$ and observe that $\lVert (1-t)f + tg\rVert_p$ is constant independent of $t \in [0,1]$ for both $p = 1$ and $p=\infty$ so that $L_1(\mathbb T)$ and $L_\infty(\mathbb T)$ are not even strictly convex.