Piecewise linear approximation of smooth functions of two variables

Abstract

Given a piecewise linear (PL) function $p$ defined on an open subset of $\R^n$, one may construct by elementary means a unique polyhedron with multiplicities $\D(p)$ in the cotangent bundle $\R^n\times \R^{n*}$ 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 $p_1,p_2,\dots$ 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.