# Purely unrectifiable metric spaces and perturbations of Lipschitz functions

##
David Bate

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

**Abstract.**

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.