This is a followup to Bill Thurston's question
about different notions of hulls.
Here I want to raise a question about the
*reflex-free hull*, which is, intuitively, the smallest
enclosing shape to an object that cannot hold water in any
orientation.
Let $S$ be a closed solid object in $\mathbb{R}^3$, and $\partial S$ its surface.
Let $H$ be a closed hemispherical neighborhood, a ball intersected
with a closed halfspace through its center.
Define a *reflex point* $p$ on $\partial S$ to be one such
that it has a neighborhood $H$ such that
(a) $H \subset S$ and (b) $H \cap \partial S = p$.
An object is *reflex-free* if it has no reflex points.
Intuitively, a reflex point could hold a drop of water in its
exterior neighborhood in some orientation.
For example, this shape is reflex-free:

The *reflex-free hull* of an object $O$ is the intersection of all
reflex-free shapes that enclose $O$.
This notion was introduced in the interesting paper cited below.
It has application to manufacturing by molten-metal casting,
and applications to architecture.They established a number of properties of the reflex-free hull,
but could not find
an algorithm to construct it.

Q1. Provide a finite algorithm to construct the reflex-free hull for a polyhedron.

They identified a number of difficulties that various ideas
for algorithms would encounter.
An algorithm that fills in cavities naively,
approaches, but never reaches, the reflex-free hull
of this example (their Fig. 7):

Q2. Is the reflex-free hull the same as Thurston's "knife hull"? (Answered by Bill Thurston below: No.)

**Reference**.
Hee-kap Ahn, Siu-Wing Cheng, Otfried Cheong, Jack Snoeyink.
"The Reflex-Free Hull."
In *Proc. 13th Canadian Conference on Computational Geometry*, 2001,
and in *International Journal of Computational Geometry and Applications*, 14(6):453-474, 2004.

- [CiteSeer][4].
- [ps for preliminary 4-page abstract][5].

(See comments for links, which are not working here for some reason...)