This is a followup to [Bill Thurston's question][1]
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:

<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
![Shape][2]
<br />

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.

> <b>Q1</b>.
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):
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
![Fig][3]
<br />

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

<b>Reference</b>.
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 link](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.259)).


  [1]: https://mathoverflow.net/questions/39229/what-can-be-said-about-the-shadow-hull-and-the-sight-hull
  [2]: https://i.sstatic.net/qHgSs.jpg
  [3]: https://i.sstatic.net/dHRcn.jpg