Pointwise differentiability of higher order for sets

TL;DR

This paper introduces two new concepts for analyzing higher order pointwise differentiability of arbitrary sets in Euclidean space, with implications for rectifiability and varifold support analysis.

Contribution

It develops two novel definitions of higher order differentiability for sets based on distance functions, and proves properties like Borel measurability and rectifiability of differentiability points.

Findings

Differentials are Borel functions.

Set of differentiability points is higher order rectifiable.

Almost all points in support intersecting a plane are strongly differentiable.

Abstract

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.