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We show that if $\Omega\subset \mathbb{R}^n$ is a bounded Lipschitz domain,and if $p\in L^\infty(\Omega)$, $p\geq 1$, then for every $f\inC^\infty(\bar\Omega)$ the following Poincar\'e--Wirtinger-type inequality holds  \begin{equation*}  \int_\Omega \big|f(x)-\langle f \rangle_{\Omega}\big|^{p(x)} \, dx  \leq C \int_\Omega\int_\Omega \frac{|\nabla f(z)|^{p(x)}}{|z-x|^{n-1}}\, dz\,dx,  \end{equation*}  where $\langle f \rangle_{\Omega}$ denotes the mean of $f$ over $\Omega$, and$C>0$ is a positive constant depending only on $\Omega$ and$\|p\|_{L^\infty(\Omega)}$. The proof we present avoids any compactnessargument and, as a particular case, also provides a complete constructive proofof the standard Poincar\'e-Wirtinger inequality in the classicalconstant-exponent context. In addition, our constructive approach offers anexplicit upper bound on the optimal Poincar\'e constant on Lipschitz domains.

A modular Poincar\'e-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents