This paper introduces a new
method for proving the infinitesimal rigidity of a broad class of polyhedra, the
caps-with-collars and their projective (Darboux) transforms, which include, as special
cases, the traditional closed convex polyhedra of Cauchy and the refined closed
convex and open convex polyhedra of Alexandrov with total curvature 2π. By
definition, a cap-with-collars consists of an oriented generalized polyhedral cap with
cylindrical polyhedral collars attached to the boundary. The spherical image of the
cap (by unit normals coherent to the orientation) must lie in some hemisphere with
the collars glued to the boundary of the cap so that their faces are parallel to the
polar vector of the hemisphere. Moreover, caps are required to satisfy a local
convexity condition, called edge-convexity, which is weaker than traditional
convexity. An edge-convex polyhedron need not have a local supporting plane
at each point. This allows great topological and morphological variety. A
cap-with-collars can have arbitrary Euler characteristic. Among the examples given
some are nonconvex; some are surfaces of genus greater than one; some are
self-intersecting surfaces; some have branch points and some have pinch
points.
