We consider the piecewise linear approximation of saddle functions of the form (f(x,y)=ax^2-by^2 ) under the (L_ ) error norm. We show that interpolating approximations are not optimal. One can get slightly smaller errors by allowing the vertices of the approximation to move away from the graph of the function.
We study piecewise linear approximation of quadratic functions de- flned on R n . Invariance properties and canonical Caley Klein metrics that help in understanding this problem can be handled in arbitrary dimensions. However, the problem of optimal approximants in the sense that their linear pieces are of maximal size by keeping a given error tolerance, is a di-cult one. We present a detailled discussion of the case n = 2, where we can partially use results from convex geometry and discrete geometry. The case n = 3 is considerably harder, and thus just a few results can be formulated so far.