Geodesic
On the rigidity of a~glued piecewise convex surface
Sbornik. Mathematics, Tome 193 (2002) no. 2, pp. 231-246

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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.) 
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     author = {P. E. Markov and O. Trejos},
     title = {On the rigidity of a~glued piecewise convex surface},
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P. E. Markov; O. Trejos. On the rigidity of a~glued piecewise convex surface. Sbornik. Mathematics, Tome 193 (2002) no. 2, pp. 231-246. http://geodesic.mathdoc.fr/item/SM_2002_193_2_a3/