Canonical decompositions of piecewise affine mappings, polyhedra-traces, and geometrical variational problems (Q950811)

The author studies the structure of piecewise affine mappings. Two classes of piecewise affine mappings are considered: mappings nondegenerating straight line segments and contractions. One shows that any piecewise affine mapping can be represented as the composition of a piecewise affine contraction and a mapping nondegenerating straight line segments. Thus, one can introduce a natural equivalence relation on piecewise affine mappings having as equivalence classes the multidimensional analogs of network classes appearing in the theory of extremal networks. They are called the polyhedron traces. The author defines the notions of volume and deformation of polyhedra traces. For the obtained volume, the author obtains a variational formula leading to an analog of the Plateau principles.