Given $f \in L^1 (\mathbb R^d)$, and $\varepsilon > 0$, define the minimal function $m_\varepsilon f$ by

$$m_\varepsilon f(x) := \inf_{B} \frac{1}{|B|} \int_B |f| ,$$

where the infimum is taken over all balls $B$ containing $x$ of radius less than or equal to $\varepsilon$, and the integral is with respect to Lebesgue measure.

Question: Is it true that for every $\varepsilon > 0$, there exists a constant $C_\varepsilon > 0$ depending only on the dimension $d$ such that for all $f \in L^1(\mathbb R^d)$,

$$\|m_\varepsilon f\|_{L^1} \geq C_\varepsilon \|f\|_{L^1}?$$

Further, is it true that the optimal constants $C_\varepsilon$ converge to $1$ as $\varepsilon \to 0$?

Remark: By the Lebesgue differentiation theorem we have $|m_\varepsilon f| \leq |f|$ a.e., and hence $\|m_\varepsilon f\|_{L^1} \leq \|f\|_{L^1}$. Thus if the conjectured result is true then we would have $m_\varepsilon f \sim_{\varepsilon} f$, with scale tending to $1$ as $\varepsilon \to 0$.