# The reflex-free hull: Construction?

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...)

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@jc: Yes, $\partial S$ is a closed surface, and $S$ is the solid whose boundary is $\partial S$. Links [4] & [5] are in what I posted, but for some reason they do not show up. If anyone knows how to fix that, please edit. Meanwhile, here they are again: citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.21.259 cccg.ca/proceedings/2001/snoeyink-68900.ps.gz –  Joseph O'Rourke Sep 20 '10 at 14:54

Consider any graph embedded in a ball, mostly trivalent but with some vertices of order 1 attached to faces of a ball, embedded in a way that each interior vertex is in the convex hull of its neighbors. Not all graphs admit such an embedding, but many do; examples can be constructed, using for example harmonic maps. For such a graph, you can hollow out of the ball a narrow tubular neighborhood whose complement is reflex-free These give examples where the knife-hull is different from the reflex-hull --- just for a $Y$ graph, the knife hull could be convex.
Here's an alternative characterization of reflexes (a reflexive definition?): a point $p$ is in a reflex for $S$ if there is a plane $P$ through $p$ and a region $R$ on the plane containing $p$ in its interior, such that the boundary of $R$ is contained in $\partial S$ and is null-homologous on $\partial S$ (bounds a region on $\partial S$). This is equivalent to the definition using the special case of half-balls, because if the shape is turned so that the plane is horizontal and the surface on $\partial S$ is below, we can look at a point where the height function has a local minimum. If it's not a strict local minimum, turn by a slight random angle to make a strict minimum, in which case we can find a half-ball.