Calculus of Variations and Geometric Measure Theory

Workshop MAR: Metric Analysis and Regularity

Purely unrectifiable metric spaces and perturbations of Lipschitz functions

David Bate (University of Warwick)

created by dimarino on 18 Sep 2018

Università di Catania


We characterise purely $n$-unrectifiable subsets of a complete metric space with finite $n$-dimensional Hausdorff measure by studying non-linear projections (i.e. 1-Lipschitz functions) into some fixed Euclidean space. We will show that a typical (in the sense of Baire category) non-linear projection maps $S$ to a set of zero $n$-dimensional Hausdorff measure. Conversely, a typical non-linear projection maps an $n$-rectifiable subset to a set of positive $n$-dimensional Hausdorff measure.

These results provide a replacement for the classical Besicovitch-Federer projection theorem, which is known to be false outside of Euclidean spaces.