Understanding an the last step in proving the Poincaré inequality for smooth functions on $\mathbb{R}^n$.

Question

I'm looking at the following: Let $u:X \to \mathbb{R}$ be an integrable and smooth function on a subspace $X \subseteq \mathbb{R}^n$ equipped with the Lebesgue measure $\lambda_n$, and let $B \subseteq X$ be an open ball. $C$ and $R$ are constants. $$ \int_B \int_B |u(x)-u(y)|\mathrm{d}y\mathrm{d}x \leq C R^{n+1} \int_B |\nabla u(x)| \mathrm{d}x $$ Supposing that this inequality is true (I've been able to derive this), how does one deduce the Poincaré inequality for smooth functions on balls in $\mathbb{R}^n$, namely $$ \frac{1}{\lambda_n(B)} \int_B |u(x)-u_B|\mathrm{d}x \leq C(n) \cdot \operatorname{diam}(B) \cdot \frac{1}{\lambda_n(B)} \int_B |\nabla u(x)| \mathrm{d}x $$ where $u_B := \frac{1}{\lambda_n(B)}\int_B u(y) \mathrm{d}y$ is the mean value of $u$ in $B$, and $C(n)$ is some positive real constant depending only on $n$?