Is there a lower bound for $\int_{B_{r}}f$ when $f$ is a positive function? Let $f$ be a positive Lebesgue integrable function on a ball $B_{r}\in\mathbb{R}^n$. I'm looking for a positive constant $c$ depending only on $f$ that satisfies
$$c\omega_nr^n\leq\int_{B_r}f$$
where $\omega_n$ is the volume of the $n$-dimensional unit ball. 
I'm also wondering if the similar result holds for a nonnegative integrable function $f$ and a nonnegative constant $c=c(f)$. Of course, $c=0$ satisfies the inequality in the this case but I need a positive constant $c$ unless $f\equiv 0$ a.e. on $B_r$. 
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I have asked these questions as I thought that the existence of such a constant will help to fill out the gap of the proof of Proposition 2.27(Alexandroff's Maximum Principle of Quasilinear Equations) in "Elliptic Partial Differential Equations" by Q. Han ad F. Lin. 
In the proof it derives an inequality
$$\int_{B_M}g^n\leq\int_{\Omega}h^n<\left(\int_{\mathbb{R}^n}g^n\right)$$
where $g\in L^n_{loc}(\mathbb{R}^n)$ and $h\in L^n(\Omega)$ are nonnegative functions and $M$ a positive constant. (The second inequality is just an assumption of the proposition.) Then the authors conclude with no further explanation that there exists a positive constant $C$ which depends only on $g$ and $h$ such that 
$$M\leq C.$$
I think that in order to derive the second inequality from the first one we need a bound which I have suggested. Thanks for your help.

Regarding the original problem you edited in: just take $C$ to be the supremum of $M$ such that the first inequality is true. Since $\int_{B_M} g^n \to \int_{\mathbb{R}^3} g^n$ as $M \to \infty$ and $\int_{\Omega} h^n < \int_{\mathbb{R}^3} g^n$, we know $\int_{B_M} g^n$ must eventually stay greater than $\int_{\Omega} h^n$, and thus the supremum must be finite.
