The Besicovitch-Federer projection theorem is false in every infinite dimensional Banach space

David Bate, Marianna Csörnyei, Bobby Wilson

We construct a purely unrectifiable set of finite ${\mathcal{H}}^1$-measure in every infinite dimensional separable Banach space $X$ whose image under every $0\ne x^{\ast }\in X^{\ast }$ has pos…