Suppose $P$ is a convex polyhedron in $\mathbb{R}^{2,1}$. Each face of $P$ comes with induced metric tensor, if the face is space-like, then it is euclidean metric; every time-like face is isometric to a convex polygon in $\mathbb{R}^{1,1}$, there might be also light-like faces with degenerate metric.
So the surface $\partial P$ comes with a piecewise constant metric of variable signature and we can talk about isometric surfaces (via piecewise isometry). The following statement is closely related to Alexandrov's uniqueness theorem.
Suppose that two convex polyhedra $P$ and $P'$ have isometric surfaces $\partial P$ and $\partial P'$. Is it true that $P$ and $P'$ are congruent?
If one reduces generality, to convex polyhedral hats (= space-like polyhedral surfaces with boundary on space-like plane), then the statement seems to be known; see "Space-like convex surfaces..." by Anatoliy Milka. I suspect that no one considered the question above, but maybe there are more partial cases --- please let me know.
