  The Guillemin boundary condition appears naturally in the study of K\"ahler
geometry of toric manifolds. In the present paper, the following Guillemin
boundary value problem is investigated \begin{align} \label{eq1} &\det D^2
u=\frac{h(x)}{\prod_{i=1}^N l_i(x)},\quad\text{in}\quad\quad P\subset\mathbb
R^n, \quad\quad \quad \quad\quad \quad \quad \quad\quad (1)\\ \label{bdy1}
&u(x)-\sum_{i=1}^N l_i(x)\ln l_i(x)\in C^\infty(\overline{P}).
\quad\quad\quad\quad \quad \quad\quad \quad \quad \quad\quad\quad (2)
  \end{align}
  Here
  \begin{equation*}
  0<h(x)\in C^\infty(\overline{P}),\quad P=\cap_{i=1}^N \{l_i(x)>0\}
  \end{equation*}
  is a simple convex polytope in $\mathbb R^n$. The solvability of (1)-(2) is
given under the necessary and sufficient condition. The key issue in the proof
is to obtain the boundary regularity of $u(x)-\displaystyle \sum_{i=1}^N
l_i(x)\ln l_i(x)$. Due to the difficulty caused by the structure of the
equation itself and the singularity of $\partial P$, we need to pay special
attention to the influence of the difference of singularity types at different
positions on $\partial P$ on the behavior of $u$ in its vicinity.
