Morrey's classical inequality implies the H & ouml;lder continuity of a function whose gradient is sufficiently integrable. Another consequence is the Hardy-type inequality lambda & Vert;ud Omega 1-n/p & Vert;infinity p <=integral Omega|Du|pdxfor any open set Omega & subne;Rn. 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.