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An umbilic point of a smooth ($C^2$) convex surface in Euclidean 3-space is a point where the principal curvatures are equal. Is there some good way to generalize this notion to convex polyhedra? See this earlier question, specially the comments by Igor Rivin and Ivan Izmestiev for the motivation behind the current question.

First, what do I mean by "good"? Here are some requirements that come to mind (which may be negotiable):

  1. It is not necessary for the notion of umbilic to be defined for all convex polyhedra but it would be enough if it is well-defined for generic polyhedra, or a class of polyhderda which is dense in the space of all convex polyhedra, and possibly open as well.

  2. Each convex polyhedron (or each polyhedron in the generic subclass) should have at least one umbilic point.

  3. It would be better if each polyhedron had at least two umbilics, or otherwise there was a notion of index associated to each umbilic so that the sum of the indices was constant for all polyhedra (in parallel with the index formula for the singularities of a line field on a sphere).

  4. It would be reassuring if there were some convex polyhedra with exactly two umbilics.

  5. (Bonus) Ultimately it would be nice if this notion were stable as a family of convex polyhedra converges to a smooth convex surface, but this can be set aside for now.

Question: Does there exist a notion of umbilic point for convex polyhedra which would satisfy the first four requirements listed above?

As was suggested by Ivan Izmestiev, maybe one should consider the Gauss image around vertices of the polyhedron. More precisely, consider the outward unit normals at the faces around a vertex. These normals form the vertices of a convex spherical polygon whose edges are geodesic arcs which join a pair of normals if and only if those normals correspond to adjacent faces of the polyhedron. So a vertex would be considered umbilic if its spherical image is somehow "circular" or relatively round (as is the case in the smooth case).

Perhaps a useful guide here could be the generalized four vertex theorem for convex (spherical) polygons; see for instance Section 21 of Igor Pak's Lecture Notes and the corresponding papers of Serge Tabachnikov cited there.

I first heard about the question in the title when Brian White asked me that in 2010. My idea back then was to consider the class of convex polyhedra with quadrilateral faces, and define an umbilic point as a vertex with degree different from $4$. It is a simple exercise to show, via Euler's formula, that all convex polyhdera with quadrilateral faces must have at least $8$ umbilics in this sense. The intuition behind this definition was the notion of "circular planar quadrilateral meshes", or "circular nets" which are discrete analogues of principal curvature lines. See for instance the book of Bobenko and Suris. This then was a purely topological candidate, and I did not get much chance to think about it more since then. Perhaps it might be more fruitful to consider geometric alternatives mentioned above, or some kind of a hybrid.

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    $\begingroup$ I see a problem with the degree four vertices. It would be natural to say a degree four vertex is umbilic if its neighborhood is inscribed into a round cone. But then a generic octahedron has no umbilic vertices. $\endgroup$ – Ivan Izmestiev Oct 17 '17 at 5:18

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