Suppose you are given a 3-ball $B$ in $\mathbb{R}^3$ that is bounded by a PL sphere, a triangulation $T$ of $B$ by Euclidean tetrahedra. Is that triangulation necessarily shellable?


I know that if $T$ can be lifted to a convex hypersurface in $\mathbb{R}^4$, then it is shellable; this applies if $T$ comes from a Delaunay triangulation. But I believe most triangulations are not liftable.


I suspect the answer is "no", that $T$ is not necessarily shellable. The first obstruction I found was the existence of a knotted arc made out of two edges of the triangulation; this obviously cannot happen for triangulations of the kind that I described.



Is every triangulation of a Euclidean ball by convex tetrahedra shellable?
I haven't had time to check through the details of the construction, but the example B\_3\_9\_18 found in the proof of Theorem 2 [here](http://www.eg-models.de/models/Simplicial_Manifolds/2003.05.004/_preview.html) by Frank Lutz appears to be embeddable in 3-space.


Frank specializes in creating wickedly tiny polytopes with unexpected properties, his website may also be useful for those studying similar questions: <http://page.math.tu-berlin.de/~lutz/> (see in particular "the manifold page").


