  Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a
Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity
$m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert
polynomial $p(z)$. Experimental evidences led us to consider the idea that
$m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert
function. We give a constructive proof of this fact. Moreover, we are able to
compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of
schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of the
Hilbert function, and also the minimal Castelnuovo-Mumford regularity $m_u$ of
schemes with Hilbert function $u$. These results find applications in the study
of Hilbert schemes. They are obtained by means of minimal Hilbert functions and
of two new constructive methods which are based on the notion of
growth-height-lexicographic Borel set and called ideal graft and extended
lifting.
