  Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq1.$ For
each $p>N$ we study the optimal function $s=s_{p}$ in the pointwise inequality
\[ \left\vert v(x)\right\vert \leq s(x)\left\Vert \nabla v\right\Vert
_{L^{p}(\Omega)},\quad\forall\,(x,v)\in\overline{\Omega}\times W_{0}%
^{1,p}(\Omega). \] We show that $s_{p}\in C_{0}^{0,1-(N/p)}(\overline{\Omega})$
and that $s_{p}$ converges pointwise to the distance function to the boundary,
as $p\rightarrow\infty.$ Moreover, we prove that if $\Omega$ is convex, then
$s_{p}$ is concave and has a unique maximum point.
