Summary: Let \(\Omega\) be a bounded, smooth domain of \(\mathbb{R}^N, N \geq 1\). For each \(p > N\) we study the optimal function \(s = s_p\) in the pointwise inequality \[ | v ( x ) | \leq s(x) \|{\nabla} v \|_{L^p ({\Omega} )}, \quad \forall(x, v) \in \overline{{\Omega}} \times W_0^{1 , p}({\Omega}) . \] 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 \(p \to \infty \). Moreover, we prove that if \(\Omega\) is convex, then \(s_p\) is concave and has a unique maximum point.