On the rigidity of a glued piecewise convex surface

A rigidity test is considered for a piecewise convex surface glued from $C^2$-smooth pieces of convex surfaces with piecewise regular boundaries. The entire surface does not have to be convex; moreover, non-convex surfaces with the so-called $A$-star-shape condition are allowed. This condition is a broad generalization of the ordinary star-shape condition; it means that the points of the surface are accessible by a certain family of curves. (In the case of the ordinary star-shape condition this family of curves consists of straight rays emanating from one point inside the surface.)