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The space $\mathfrak R_{\mathrm c}(P)$ of convex realizations of a (3-dimensional, spherical) polyhedron $P$ is known to be well-behaved: it is a contractible manifold of dimension $\#\text{edges}+6$ (this result is usually attributed to Steinitz and Legendre). See below for more detailed definitions. I wonder how much of this is due to convexity, sphericity and low dimension respectively. Here I ask specifically what is known if we drop convexity:

Question: What is known about the space $\mathfrak R(P)$ of general (i.e. not necessarily convex) realizations? Has it been studied in the literature?

Typical questions that are interesting to me: is $\mathfrak R(P)$ connected/contractible after removing degenerate realizations? Since $\mathfrak R(P)$ has the structure of a (real) affine varierty, what are its irreducible components? etc

Definitions and remarks

By "polyhedron" I mean a 3-connected planar graph. Note that vertices, edges, and in particular, faces are well-defined. A "realization" thereof is then any embedding of the vertices into $\Bbb R^3$ so that vertices that belong to the same face lie in a common plane. In particular, this allows non-convex, self-intersecting and degenerate (collapsed into a plane, line or point) realizations. Some useful examples to think of are the Great icosahedron and the Bricard octahedron.

I am aware that I am asking about the realization space of a special type of matroids. While matroid realization spaces can be arbitrarily bad, I hope that my setting (3-dimensional, spherical) allows to say more.

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  • $\begingroup$ What is the realisation space of Schönhardt polyhedron ? $\endgroup$
    – Dima Pasechnik
    Commented Apr 24 at 9:45
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    $\begingroup$ "vertices that belong to the same face lie in a common plane" is not a restriction if all the faces are simplicial, so it's not clear to me how your embeddings are defined. I guess you might want to specify that the geometry is not broken, in the sense that faces intersect as they should (e.g. interiors of triangular faces don't intersect, etc) $\endgroup$
    – Dima Pasechnik
    Commented Apr 24 at 9:56
  • $\begingroup$ Wikipedia's discussion of the definition of a polyhedron lacks mathematical precision. What is a polyhedron for the purpose of your question? $\endgroup$
    – Moishe Kohan
    Commented Apr 24 at 13:35
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    $\begingroup$ In my answer I took the question-asker at their word that by "not necessarily convex realization of a polyhedron" they just mean an assignment of coordinates to the vertices so that vertices on the same face are coplanar, with no further conditions. $\endgroup$
    – Sam Hopkins
    Commented Apr 24 at 13:39
  • $\begingroup$ @DimaPasechnik That is correct, if the polyhedron is simplicial, then its realization space is very simple: any embedding of the vertices will do. I do explicitly not want to impose any constraints beyond coplanarity. I am however more interested in realization spaces of polyhedra that are not simple/simplicial. $\endgroup$
    – M. Winter
    Commented Apr 24 at 15:15

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I'm not sure if this is useful for you, but here is one way to describe the realization space. You can represent a realization of your polyhedron as a matrix whose columns give the coordinates of the vertices. Then the requirement that certain vertices be coplanar amounts to certain of the maximal minors of this matrix vanishing (to go from affine dependence to linear dependence we may need to increase the dimension by one, say by appending a row of all $1$'s at the bottom of our matrix). In other words, what we get is essentially a cell in the matroid stratification of the Grassmannian.* In general it is known that cells in the matroid stratification of the Grassmannian can be arbitrarily badly behaved, e.g., have the singularities of any kind of algebraic variety. But of course your cells are a little special, so maybe more can be said.

*Maybe there is an issue if you want to allow really degenerate embeddings such as the case where your polyhedron collapses to a single point, because a point in the Grassmannian should be represented by a full rank matrix. There is also the issue of real versus complex entries (the complex Grassmannian is definitely more studied), but that should not make a huge difference.

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  • $\begingroup$ In my question I focus on dimension three exactly because I know that realization spaces of higher-dimensional (convex) polytopes can be arbitrarily bad. Since the realization space of convex 3-dimensional polytopes is so well-behaved, I wondered how much of this comes from the convexity, or whether it's all in the sphericity and dimension three. $\endgroup$
    – M. Winter
    Commented Apr 23 at 21:13
  • $\begingroup$ Fair enough. I still think you may find what you are looking for in the matroid theory literature, so I'll leave this up so that matroids are mentioned on this page. $\endgroup$
    – Sam Hopkins
    Commented Apr 23 at 21:17
  • $\begingroup$ I agree and I thought about bringing up matroid in the question. Is there any specific terminology for matroids that corresponds to this "sperical structure" that comes with a polyhedron? $\endgroup$
    – M. Winter
    Commented Apr 23 at 21:36
  • $\begingroup$ @M.Winter once you consider non-convex realizations, you are just specifying certain minors to be zero, so there does not seem to me to be much special beyond the fact that you are in dimension three. $\endgroup$
    – Sam Hopkins
    Commented Apr 23 at 21:49

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