Dilatations of graphs and Taylor's formula: some results about convergence.

Abstract

The graph of a function $f$ is subjected to non-homogeneous dilatations around the point $(x_0;f(x_0))$, related to the Taylor's expansion of $f$ at $x_0$. Some questions about convergence are considered. In particular the dilated images of the graph are proved to behave nicely with respect to a certain varifold-like convergence. Further and stronger results are shown to hold in such a context, by suitably reinforcing the assumptions.