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

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$\Bbb{T} = \Bbb{R}\setminus\Bbb{Z}$? Do you mean $\Bbb{T} = \Bbb{R}/\Bbb{Z}$ (circle group)? – Stahl Mar 3 '13 at 7:43
@Stahl: Yes. I am sorry for the confusion – cimt Mar 3 '13 at 7:59

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up vote 5 down vote accepted

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.

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