  Morrey's classical inequality implies the H\"older continuity of a function
whose gradient is sufficiently integrable. Another consequence is the
Hardy-type inequality $$
\lambda\biggl\|\frac{u}{d_\Omega^{1-n/p}}\biggr\|_{\infty}^p\le \int_\Omega
|Du|^p \,dx $$ for any open set $\Omega\subsetneq \mathbb{R}^n$. This
inequality is valid for functions supported in $\Omega$ and with $\lambda$ a
positive constant independent of $u$. The crucial hypothesis is that the
exponent $p$ exceeds the dimension $n$. This paper aims to develop a basic
theory for this inequality and the associated variational problem. In
particular, we study the relationship between the geometry of $\Omega$, sharp
constants, and the existence of a nontrivial $u$ which saturates the
inequality.
