  Let $I$ be a monomial ideal in a polynomial ring $A=K[x_1,...,x_n]$. We call
a monomial ideal $J$ to be a minimal monomial reduction ideal of $I$ if there
exists no proper monomial ideal $L \subset J$ such that $L$ is a reduction
ideal of $I$. We prove that there exists a unique minimal monomial reduction
ideal $J$ of $I$ and we show that the maximum degree of a monomial generator of
$J$ determines the slope $p$ of the linear function $\reg(I^t)=pt+c$ for $t\gg
0$. We determine the structure of the reduced fiber ring
$\mathcal{F}(J)_{\red}$ of $J$ and show that $\mathcal{F}(J)_{\red}$ is
isomorphic to the inverse limit of an inverse system of semigroup rings
determined by convex geometric properties of $J$.
