\magnification=1200
\hsize=4in
\overfullrule=0pt
\input amssym
%\def\frac2{{#1\over #2}}
\def\emph#1{{\it #1}}
\def\em{\it}
\nopagenumbers
\noindent
%
%
{\bf Hidefumi Ohsugi and Takayuki Hibi}
%
%
\medskip
\noindent
%
%
{\bf The $h$-Vector of a Gorenstein Toric Ring of a Compressed Polytope}
%
%
\vskip 5mm
\noindent
%
%
%
%
A compressed polytope is an integral convex polytope
all of whose pulling triangulations are unimodular.
A $(q - 1)$-simplex $\Sigma$ each of whose vertices is
a vertex of a convex polytope
${\cal P}$ is said to be a special simplex in ${\cal P}$
if each facet of ${\cal P}$ contains exactly
$q - 1$ of the vertices of $\Sigma$.
It will be proved that there is a special simplex
in a compressed polytope ${\cal P}$ if (and only if)
its toric ring $K[{\cal P}]$
is Gorenstein. In consequence it follows that
the $h$-vector of a Gorenstein toric ring $K[{\cal P}]$
is unimodal if ${\cal P}$ is compressed.
\bye