# Poincaré inequality

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Let $f\in W^1_p(\mathbb R^n)$, $1\leqslant p < n$ and $p^* = \frac{np}{n-p}$ then the following inequality holds $$\label{eq:1} \Bigl(\int\limits_{B}|f(x)-f_B|^{p^*}\,dx\Bigr)^{\frac{1}{p^*}} \leqslant C\Bigl(\int\limits_{B}|\nabla f(x)|^{p}\,dx\Bigr)^{\frac{1}{p}}$$ for any balls $B \subset \mathbb R^n$, and constant $C$ depends only on $n$ and $p$.