An optimal Poincaré inequality in $L^1$ for convex domains

Abstract

For convex domains $\Omega \subset \mathbb {R}^n$ with diameter $d$ we prove \[ \|u\|_{L^1(\omega )} \le \frac {d}{2} \|\nabla u\|_{L^1(\omega )} \] for any $u$ with zero mean value on $\omega$. We also show that the constant $1/2$ in this inequality is optimal.