  The antiprism triangulation provides a natural way to subdivide a simplicial
complex $\Delta$, similar to barycentric subdivision, which appeared
independently in combinatorial algebraic topology and computer science. It can
be defined as the simplicial complex of chains of multi-pointed faces of
$\Delta$, from a combinatorial point of view, and by successively applying the
antiprism construction, or balanced stellar subdivisions, on the faces of
$\Delta$, from a geometric point of view.
  This paper studies enumerative invariants associated to this triangulation,
such as the transformation of the $h$-vector of $\Delta$ under antiprism
triangulation, and algebraic properties of its Stanley--Reisner ring. Among
other results, it is shown that the $h$-polynomial of the antiprism
triangulation of a simplex is real-rooted and that the antiprism triangulation
of $\Delta$ has the almost strong Lefschetz property over ${\mathbb R}$ for
every shellable complex $\Delta$. Several related open problems are discussed.
