  The well known duality between the Sobolev inequality and the
Hardy-Littlewood-Sobolev inequality suggests that the Nash inequality could
also have an interesting dual form, even though the Nash inequality relates
three norms instead of two. We provide such a dual form here with sharp
constants. This dual inequality relates the $L^2$ norm to the infimal
convolution of the $L^\infty $ and $H^{-1}$ norms. The computation of this
infimal convolution is a minimization problem, which we solve explicitly, thus
providing a new proof of the sharp Nash inequality itself. This proof, via
duality, also yields the sharp form of some new, weighted generalizations of
the Nash inequality as well as the dual of these weighted variants.
