If I understand correctly, reflex-free is equivalent to the property that the orthogonal projection to any line has no inside local minimum, that is, there is no point where the function is a local minimum on the boundary but not in the solid. For a smooth surface, this means both principle directions can't be curving in the concave sense.

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 $\partial S$). Could this characterization find the reflex hull in one go?

I wonder whether a variation of minimal surfaces would give a reasonable procedure to find the reflex-free hull. For a minimal surface, the two principal curvatures balance---you can think of it like rubber bands in perpendicular directions in a tug of war, trying to pull surface in opposite directions but coming to a draw. Now imagine a surface whose molecules align to pull very hard in the convex directions, and only weakly in the concave directions. To those who have ... if you take the limit of inward-pulled surfaces, as insiders grow stronger and outsiders weaker, seems like a candidate reflex-free hull.

Minimal surfaces can be difficult to analyze because there are often local minima that are not global minima, and unstable critical points of the area function as well. However, with inward-pulled surfaces, I have trouble thinking of examples where such a phenomena occurs. If there is only one local minimum, then computation might be relatively efficient.

If I understand correctly, reflex-free is equivalent to the property that the orthogonal projection to any line has no inside local minimum, that is, there is no point where the function is a local minimum on the boundary but not in the solid. For a smooth surface, this means both principle directions can't be curving in the concave sense.

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.

I wonder whether a variation of minimal surfaces would give a reasonable procedure to find the reflex-free hull. For a minimal surface, the two principal curvatures balance---you can think of it like rubber bands in perpendicular directions in a tug of war, trying to pull surface in opposite directions but coming to a draw. Now imagine a surface whose molecules align to pull very hard in the convex directions, and only weakly in the concave directions. To those who have ... if you take the limit of inward-pulled surfaces, as insiders grow stronger and outsiders weaker, seems like a candidate reflex-free hull.

Minimal surfaces can be difficult to analyze because there are often local minima that are not global minima, and unstable critical points of the area function as well. However, with inward-pulled surfaces, I have trouble thinking of examples where such a phenomena occurs. If there is only one local minimum, then computation might be relatively efficient.