A polyhedron is the intersection of a finite collection of halfspaces. These halfspaces are not assumed to be linear, i.e. their bounding hyperplanes are not assumed to contain the origin. The support Supp(M) of a collection M of polyhedra is the union of the polyhedra in M. I can prove the following theorem:
Theorem. Let M be a finite set of n-dimensional polyhedra in Rn. Suppose:
(i) The interior of Supp(M) is path-connected; and
(ii) For every x in the boundary of Supp(M), there exists a closed halfspace H+ bounded by a hyperplane H such that x is in H, and such that H+ contains every P in M such that x is in P.
Then Supp(M) is convex.
(Acknowledgment: I proved a characterization of coarsenings of a given polyhedral complex and Ezra Miller remarked that part of my argument amounted to some sort of local criterion for convexity. The theorem above is that criterion.)
The point here is that you only need to check, at each point x of the boundary, that Supp(M) looks sufficiently like a convex set near x, and (ii) says exactly what "sufficiently like a convex set" means in this case.
The question is:
Is this a special case of some general theorem that says that convexity is somehow a local condition?
I suspect that I'm asking for a reference to something known. One convexity person that I asked about felt that it is "highly likely..., that this result is a special case of a result in functional analysis, once properly understood." The same person suggested that there might be a connection to the theory of tight manifolds in topology. For that reason I have added the tags fa.functional-analysis and gt.geometric-topology. My apologies if these tags turn out not to be appropriate.