Piecewise linear approximation of smooth functions of two variables

2013/05/09 by Fu, Joseph H. G., Scott, Ryan C.
#26B25 #53A05 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.2220

Abstract

Given a piecewise linear (PL) function p defined on an open subset of \Rⁿ, one may construct by elementary means a unique polyhedron with multiplicities \D(p) in the cotangent bundle \Rⁿ× \R^n* representing the graph of the differential of p. Restricting to dimension 2, we show that any smooth function f(x,y) may be approximated by a sequence p₁,p₂,… of PL functions such that the areas of the \D(p_i) are locally dominated by the area of the graph of df times a universal constant.

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