Annals of Global Analysis and Geometry

, Volume 55, Issue 3, pp 591–621 | Cite as

Pointwise differentiability of higher order for sets

  • Ulrich MenneEmail author


The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that differentials are Borel functions, higher-order rectifiability of the set of differentiability points, and a Rademacher result. One concept is characterised by a limit procedure involving inhomogeneously dilated sets. The original motivation to formulate the concepts stems from studying the support of stationary integral varifolds. In particular, strong pointwise differentiability of every positive integer order is shown at almost all points of the intersection of the support with a given plane.


Higher-order pointwise differentiability Rectifiability Rademacher–Stepanov type theorem Stationary integral varifold 

Mathematics Subject Classification

Primary: 51M05 Secondary: 26B05 49Q20 



The author would like to thank Mario Santilli for reading part of the manuscript and for bringing a series of papers of Isakov to his attention, Dr. Sławomir Kolasiński for helping him to become acquainted with some of these results available only in Russian, and Dr. Yangqin Fang for pointing him to [11]. The initial version of this paper (see was written while the author worked at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and the University of Potsdam. The subsequent revision was made while the author worked at the University of Leipzig and the Max Planck Institute for Mathematics in the Sciences.


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Authors and Affiliations

  1. 1.Department of MathematicsNational Taiwan Normal UniversityTaipei CityTaiwan, ROC

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