Series expansion for L^p Hardy inequalities

We consider a general class of sharp $L^p$ Hardy inequalities in ${\mathbb{R}}^N$ involving distance from a surface of general codimension $1\le k\le N$. We show that we can succesively improve them by adding to the right hand side a lower order term with optimal weight and best constant. This leads to an infinite series improvement of $L^p$ Hardy inequalities.