Let $I$ be a monomial ideal in a polynomial ring $A=
K[x_1,\ldots,x_n]$. We call a monomial ideal $J$ 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 $J$.