Calculus of Variations and Geometric Measure Theory

Purely unrectifiable metric spaces and perturbations of Lipschitz functions

David Bate (University of Warwick)

created by gelli on 09 Apr 2018
modified on 22 Apr 2018

16 may 2018 -- 17:00   [open in google calendar]

Sala Seminari Dipartimento di Matematica di Pisa


We give characterisations of purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of Lipschitz functions $f\colon X \to R^m$. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, a typical $f$ (with respect to the supremum norm) has ${\mathcal H}^n(f(S))=0$. Conversely, if $E\subset X$ is $n$-rectifiable with ${\mathcal H}^n(E)>0$, a typical $f$ has ${\mathcal H}^n(f(E))>0$. These results provide a replacement for the Besicovitch-Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.