A geometrical version of Hardy’s inequality for \stackrel{∘}𝑊^{1,𝑝}(Ω)

The aim of this article is to prove a Hardy-type inequality, concerning functions in W ∘ 1 , p ( Ω ) \stackrel {\circ }{\textrm {W}}{\!}^{1,p}(\Omega ) for some domain Ω ⊂ R n \Omega \subset R^n , involving the volume of Ω \Omega and the distance to the boundary of Ω \Omega . The inequality is a generalization of a recently proved inequality by M. Hoffmann–Ostenhof, T. Hoffmann–Ostenhof and A. Laptev (2002), which dealt with the special case p = 2 p=2 .