# 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.

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