*Note: We view the sphere $S^1$ as $[0,1]$ with the endpoints identified, and equip it with its usual addition structure, and Lebesgue measure.*


**Question:** Does there exist an absolute constant $C > 0$ such that for all $L^1$ functions $f: S^1 \to \mathbb R$,


$$\sup\_{t \in S^1} \int\_{S^1} |f(x + t) - f(x)| \, dx \geq C\int\_{S^1} \left|f(x) - \left(\int\_{S\_1} f(y) \, dy\right) \right| \,dx\text{?}$$



The maximal difference between a function and translates of itself
Using the condition $\int\_{S^1}dx=1$ and Jensen's inequality, we have
$$\sup\_{t\in S^1} \int\_{S^1}dx\,|f(x + t)-f(x)|
\ge\int\_{S^1}dt\, \int\_{S^1}dx\,|f(x + t)-f(x)| \\
=\int\_{S^1}dx\, \int\_{S^1}dt\,|f(x)-f(x + t)|
\ge\int\_{S^1}dx\, \Big|f(x)-\int\_{S^1}dt\,f(x + t)\Big| \\ 
=\int\_{S^1}dx\, \Big|f(x)-\int\_{S^1}dy\,f(y)\Big|. $$
So, your conjectured inequality holds with $C=1$. $\quad\Box$


