Do polyhedral wormholes make sense? Visser defines some class of wormholes with polyhedral mouthes, as a limit of smoothed polyhedrons as the radius of the edges go to zero. Does this limit actually make sense, as an actual spacetime? That is, can I remove two polyhedrons from Minkowski space and identify them in a way that makes sense as a Lorentzian manifold? I've got some doubts due to the following issues : 


*

*Gluing together manifolds require a collared neighbourhood around the manifold edges, which I'm not sure exist if the edge isn't smooth (this isn't even a manifold with boundaries, it's a manifold with corners)

*If such a gluing can be performed, I don't think the resulting manifold is even $C^1$

*It's a theorem that metrics of the Geroch-Traschen class cannot be generated by linear densities, only by surface densities, meaning that it could only be described by Colombeaux algebras in GR


So can such a spacetime be constructed, or should it just be considered by some very small radii of the edges and corners?
 A: Yes, such wormholes do make sense. (Subject to the usual caveats about violation of energy conditions).
The type of singularity for this class of wormholes is the conical singularity of cosmic string (here, with negative tension). For example, for the cube wormhole, by circling an edge (over both universes) we observe the total angle $3 \pi$ (angle excess of $\pi$).  By increasing the number of faces the angle excess (and thus negative linear density) could be made much smaller, potentially so that the 'weak field' limit becomes applicable. 
The subject of cosmic strings has a large body of literature. For example, see a rather old review:

Hindmarsh, M.B., & Kibble, T. W. B. (1995). Cosmic strings. Reports on Progress in Physics, 58(5), 477, doi, arXiv.

The bibliography has 300+ papers. The authors of the review do not particularly concern themselves with smoothness classes, but do obtain $\delta$-like energy-momentum tensor.
But more formal treatment of conical singularity (with Colombeau's generalized functions) is given in:

Clarke, C. J. S., Vickers, J. A., & Wilson, J. P. (1996). Generalized functions and distributional curvature of cosmic strings. Classical and Quantum Gravity, 13(9), 2485, doi, arXiv.

Additionally, such spacetimes are (sometimes implicitly) assumed to have an underlying microscopic structure, so while from a macroscopic point of view we observe distributional structure of, say, Ricci tensor there exists a scale at which the metric, connection, and curvature tensors are smooth. While in general for line distributions of energy-momentum, the limit of zero-width is poorly defined it does exist in the case of conical singularity:

Hindmarsh, M., & Wray, A. (1990). Gravitational effects of line sources and the zero-width limit. Physics Letters B, 251(4), 498-502, doi.

