Geodesic
Canonical decompositions of piecewise affine mappings, polyhedra-traces, and geometrical variational problems
Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 1, pp. 57-94
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In this paper, canonical decompositions of arbitrary piecewise affine mappings are constructed. Then the equivalence of these mappings is introduced and the concept of polyhedron-trace is defined as an equivalence class. Finally, the concepts of the volume and the deformation of polyhedra-traces are introduced, the continuity of the volume is proved, and the formula of first variation is obtained. These concepts give an analog of the Plateau principles. 
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N. S. Gusev. Canonical decompositions of piecewise affine mappings, polyhedra-traces, and geometrical variational problems. Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 1, pp. 57-94. http://geodesic.mathdoc.fr/item/FPM_2006_12_1_a1/

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