  In 1980 Provan and Billera defined the notion of weak $k$-decomposability for
pure simplicial complexes. They showed the diameter of a weakly
$k$-decomposable simplicial complex $\Delta$ is bounded above by a polynomial
function of the number of $k$-faces in $\Delta$ and its dimension. For weakly
0-decomposable complexes, this bound is linear in the number of vertices and
the dimension. In this paper we exhibit the first examples of non-weakly
0-decomposable simplicial polytopes.
