Cone unrectifiable sets and non-differentiability of Lipschitz functions

We provide sufficient conditions for a set E ⊂ ℝ ⁿ to be a non-universal differentiability set, i.e., to be contained in the set of points of non-differentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of non-differentiability of Lipschitz self-maps of ℝ ⁿ given by Alberti, Csörnyei and Preiss, which eventually led to the result of Jones and Csörnyei that for every Lebesgue null set E in ℝ ⁿ there is a Lipschitz map f: ℝ ⁿ → ℝ ⁿ not differentiable at any point of E, even though for n > 1 and for Lipschitz functions from ℝ ⁿ to ℝ there exist Lebesgue null universal differentiability sets. Among other results, we show that the new class of Lebesgue null sets introduced here contains all uniformly purely unrectifiable sets and gives a quantified version of the result about non-differentiability in directions outside the decomposability bundle with respect to a Radon measure.