ABSTRACT. We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is constructed explicitly.

We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.

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 nondifferentiability of a real-valued Lipschitz function. These conditions are motivated by a description of the ideal generated by sets of nondifferentiability 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.

We show that every finite-dimensional Euclidean space contains compact universal differentiability sets of upper Minkowski dimension one. In other words, there are compact sets S of upper Minkowski dimension one such that every Lipschitz function defined on the whole space is differentiable inside S. Such sets are constructed explicitly.