  In this paper we study some improvements of the classical Hardy inequality.
We add to the right hand side of the inequality a term which depends on some
Lorentz norms of $u$ or of its gradient and we find the best values of the
constants for remaining terms. In both cases we show that the problem of
finding the optimal value of the constant can be reduced to a spherically
symmetric situation. This result is new when the right hand side is a Lorentz
norm of the gradient.
