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Pointwise differentiability of higher order for sets

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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.

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Notes

  1. Whenever A is a set, \(\mathbf {1}_{A}\) denotes the identity map of A, see [14, p. 669].

  2. Equivalently, the topology on \(\mathbf {G}({n},{m})\) is characterised by the requirement that \(\mathbf {G}({n},{m})\) becomes a homogeneous space through the canonical transitive left action of the orthogonal group \({\mathbf {O}}(n)\) on \(\mathbf {G}({n},{m})\), see [14, 2.7.1, 3.2.28 (2) (4)].

  3. The symbol \(\mathbf {B}(a,r)\) denotes the closed ball with centre a and radius r, see [14, 2.8.1].

  4. The Russian original is [36].

  5. The symbol \(\mathscr {L}^{m}\) denotes the \({m}\) dimensional Lebesgue measure, see [14, 2.6.5].

  6. The Russian original is [18].

  7. The tangent cone \({{\,\mathrm{Tan}\,}}(A,a)\) consists of all \(v \in \mathbf {R}^{n}\) such that for \(\varepsilon > 0\) there exist \(x \in A\) and \(0< r < \infty \) such that \(|x-a| < \varepsilon \) and \(|r(x-a)-v| < \varepsilon \), see [14, 3.1.21]. In set-valued analysis, this cone is called “contingent cone” of A at a, see [2, 4.1.1].

  8. The symbol \(\mathscr {H}^{m}\) denotes the \({m}\) dimensional Hausdorff measure, see [14, 2.10.2].

  9. A subset of \(\mathbf {R}^{n}\) is called countably \({m}\) rectifiable if and only if it can be covered by the union of a countable family of Lipschitzian images of subsets of \(\mathbf {R}^{m}\), see [14, 3.2.14 (2)].

  10. The map f is called of class k if and only if its domain is open and it is k times continuously differentiable, see [14, 3.1.11].

  11. If V and W are vectorspaces, then \(\bigodot ^0 (V,W)=W\) and \(\bigodot ^i (V,W)\) is the vectorspace of all symmetric i linear maps from \(V^i\) into W whenever i is a positive integer, see [14, 1.10.1].

  12. If V and W are vectorspaces, i is a positive integer, and \(\phi \in \bigodot ^i (V,W)\), then

    $$\begin{aligned} \langle v^i/i!, \phi \rangle = i!^{-1} \phi ( v, \ldots , v ) \quad \text {for }v \in V, \end{aligned}$$

    where the power \(v^i\) is computed in \(\bigodot _*V\), see [14, 1.9.1, 1.10.1, 1.10.4]. Similarly, if \(i = 0\) and \(\phi \in \bigodot ^i ( V,W )\), then

    $$\begin{aligned} \langle v^i/i!, \phi \rangle = \phi \quad \text {for }v \in V. \end{aligned}$$

  13. The k jet of f at a is the polynomial function \(P : X \rightarrow Y\) of degree at most k satisfying the equation \(P (x) = \sum _{i=0}^k \langle (x-a)^i/i!, {{\,\mathrm{D}\,}}^i f(a) \rangle \) for \(x \in X\), see [14, 3.1.11].

  14. If g is a map between metric spaces, then \({{\,\mathrm{Lip}\,}}g\) is its Lipschitz constant, see [14, 2.2.7].

  15. The closure of a set A is denoted \({{\,\mathrm{\mathrm{Clos}\,}\,}}A\), see [14, p. 669].

  16. The symbol \(\mathbf {U}(a,r)\) denotes the open ball with centre a and radius r, see [14, 2.8.1].

  17. The \({m}\) dimensional density of a measure \(\phi \) over \(\mathbf {R}^{n}\) at a equals

    $$\begin{aligned} \varvec{\Theta }^{m}( \phi , a ) = \lim _{r \rightarrow 0+} \frac{{\phi }\,{\mathbf {B}(a,r)}}{\varvec{\alpha }({m})r^{m}}, \end{aligned}$$

    where \(\varvec{\alpha }({m})= {\mathscr {L}^{m}}\,{\mathbf {B}(0,1)}\) if \({m}> 0\) and \(\varvec{\alpha }(0) = 1\), see [14, 2.7.16 (1), 2.10.19].

  18. The term “univalent” is also known as “injective”.

  19. Anticipating the results of this paper and its logical sequel [37], we remark that—employing the terminology of approximate differentiation from [37, 3.8, 3.19]—the following proposition may be deduced from [37, 4.1, 4.3, 4.11] and 3.10, 3.11 (4): Whenever\(a \in {\mathbf {R}}^n\), \(A \subset {\mathbf {R}}^n\), kis a positive integer,\(0 \le \alpha \le 1\), \(\gamma = k\)if\(\alpha = 0\)and\(\gamma = ( k, \alpha )\)if\(\alpha > 0\), A is approximately differentiable of order 1 ata, \(T = {{\,\mathrm{\mathrm{ap}\,}\,}}{{\,\mathrm{Tan}\,}}(A,a)\), and\(m = \dim T \ge 1\), the following two conditions are equivalent:

    1. (1)

      The setAis approximately differentiable of order \((k,\alpha )\)ata.

    2. (2)

      There exists a subsetBof\({\mathbf {R}}^n\)that is pointwise differentiable of order\(\gamma \)ataand satisfies the conditions\({{\,\mathrm{Tan}\,}}(B,a) = T\)and\(\varvec{\Theta }^{m}( \mathscr {H}^m {{\,\mathrm{\llcorner }\,}}A{{\,\mathrm{\sim }\,}}B, a) = 0\).

    In this case, \({{\,\mathrm{\mathrm{ap}\,}\,}}{{\,\mathrm{D}\,}}^i A (a) = {{\,\mathrm{\mathrm{pt}\,}\,}}{{\,\mathrm{D}\,}}^i B (a,T)\)for\(i = 2, \ldots , k\). In (2), one may require \(B \subset A\).

  20. As the proof of [17, Theorem 2] is omitted in that reference as “completely analogous” to [17, Theorem 1], the reader may find it helpful to notice that the presently needed case of [17, Theorem 2] is in fact simpler than [17, Theorem 1] provided one refers to [40, VI.2.2.2, VI.2.3.1–3] instead of [14, 3.1.14] for the Whitney type extension theorem.

  21. Whenever X and Y are sets \(Y^X\) denotes set of maps from X into Y, see [14, p. 669].

  22. If \(A \subset \mathbf {R}\) and \(f : A \rightarrow \mathbf {R}\) then f is of class \(\infty \) relative to A if and only if there exist an open subset U of \(\mathbf {R}\) and \(g : U \rightarrow \mathbf {R}\) of class \(\infty \) with \(A \subset U\) and \(f=g|A\), see [14, 3.1.22].

  23. If g maps a subset of \(\mathbf {R}\) into \(\mathbf {R}\) and g is k times differentiable at x, then \(g^{(k)}(x) \in \mathbf {R}\) denotes the k-th derivative of g at x, see [14, 3.1.11].

  24. By definition \(\varvec{\alpha }({m})= {\mathscr {L}^{m}}\,{\mathbf {B}(0,1)}\), see [14, 2.7.16 (1)].

  25. The adjoint linear map \(\mathbf {p}^*: \mathbf {R}^{m}\rightarrow \mathbf {R}^{n}\) associated with \(\mathbf {p}\) satisfies \(\mathbf {p}^*(x) = (x_1, \ldots , x_{m}, 0 ) \in \mathbf {R}^{n}\) for \(x = (x_1, \ldots , x_{m}) \in \mathbf {R}^{m}\), see [14, 1.7.4].

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Acknowledgements

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 https://arxiv.org/abs/1603.08587v1) 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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    Ulrich Menne

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A Items employed from Federer’s treatise

A Items employed from Federer’s treatise

For the convenience of the reader, Table 1 provides a brief list of the results employed from [14]. Items which merely provide background are not listed.

Table 1 Items employed from [14]
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Menne, U. Pointwise differentiability of higher order for sets. Ann Glob Anal Geom 55, 591–621 (2019). https://doi.org/10.1007/s10455-018-9642-0

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