The research presented here developed from rather mysterious observations, originally made by the authors independently and in different circumstances, that Lebesgue null sets may have uniquely defined tangent directions that are still seen even if the set is much enlarged (but still kept Lebesgue null). This phenomenon appeared, for example, in the rank-one property of derivatives of BV functions and, perhaps in its most striking form, in attempts to decide whether Rademacher's theorem on differentiability of Lipschitz functions may be strengthened or not.We describe the non-differentiability sets of Lipschitz functions on R n and use this description to explain the development of the ideas and various approaches to the definition of the tangent fields to null sets. We also indicate connections to other current results, including results related to the study of structure of sets of small measure, and present some of the main remaining open problems. (2000). Primary 26B05; Secondary 28A75.
Mathematics Sub ject Classification
Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets which contain, for every x ∈ R n , a punctured hyperplane H \ {x} through x. With similar methods we also construct a Borel subset of R n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.2010 Mathematics Subject Classification. 28A78, 54E52.
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