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

Abstract

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 positive Lebesgue measure. This demonstrates completely the failure of the Besicovitch-Federer projection theorem in infinite dimensional Banach spaces.