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The vertex scaling for piecewise linear metrics on polyhedral surfaces wasintroduced by Luo, who proved the local rigidity by establishing a variationalprinciple and conjectured the global rigidity. Luo's conjecture was solved byBobenko-Pinkall-Springborn, who also introduced the vertex scaling forpiecewise hyperbolic metrics and proved its global rigidity.Bobenko-Pinkall-Spingborn's proof is based on their observation of theconnection of vertex scaling and the geometry of polyhedra in $3$-dimensionalhyperbolic space and the concavity of the volume of ideal and hyper-idealtetrahedra. In this paper, we give an elementary and short variational proof ofthe global rigidity of vertex scaling without involving $3$-dimensionalhyperbolic geometry. The method is based on continuity of eigenvalues ofmatrices and the extension of convex functions.

A new proof for global rigidity of vertex scaling on polyhedral surfaces