A geometric second-order-rectifiable stratification for closed subsets of Euclidean space

TL;DR

This paper introduces a new geometric framework for stratifying closed subsets of Euclidean space, proving that each stratum is second-order rectifiable and a Borel set, extending known results beyond convex sets.

Contribution

It establishes that the $m$-th stratum of a closed set is second-order rectifiable and Borel, providing a new criterion that generalizes previous results for convex sets and sets of positive reach.

Findings

Each stratum is second-order rectifiable of dimension m.

The m-th stratum is a Borel set.

The result extends known properties from convex sets to more general closed sets.

Abstract

Defining the $m$-th stratum of a closed subset of an $n$ dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least $n-m$ linearly independent directions, we establish that the $m$-th stratum is second-order rectifiable of dimension $m$ and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a new criterion for second-order rectifiability.

Equations

$$B=A\cap \{a:\text{Dis(A,a) contains at least n−m linearly independent vectors}\}.$$

$$\underset{y\to x}{\mathrm{limsup}}|y-x{|}^{-2}\mathrm{dist}(f(y)-f(x),U)<\infty \text{whenever x∈f−1[{z}]}.$$

$$\underset{y\to x}{\mathrm{limsup}}|y-x{|}^{-2}\mathrm{dist}({\mathbf{\xi }}_A(y)-a,U)<\infty .$$

$$r[A]=\{y:\text{(x,y)∈r for some x∈A}\}.$$

$$\textstyle\mathscr{H}^{m}\big{(}\mathbf{R}^{\nu}\cap\{z\operatorname{:}\mathscr{H}^{n-m}(f^{-1}[\{z\}])>0\}\operatorname{\sim}\bigcup\{f[P]\operatorname{:}P\in G\}\big{)}=0.$$

$${\mathbf{R}}^{\nu }\cap \{z:{\mathcal{H}}^{n-m}(f^{-1}[\{z\}])>0\}={\bigcup }_{i=1}^{\infty }{Z}_i.$$

$$f[P]\cap f[Q]\ne \varnothing$$

$${\mathcal{H}}^m(P)>0,\text{f∣P is univalent},\text{(f∣P)−1 is Lipschitzian},$$

$$P$$

$$\textstyle\mathscr{H}^{m}\big{(}Z_{i}\operatorname{\sim}\bigcup\{f[P]\operatorname{:}P\in G_{i}\}\big{)}=0\quad\text{for every positive integer $i$}.$$

$$\textstyle B=S\cap\big{\{}w\operatorname{:}\|\bigwedge_{m}\operatorname{D}F(w)\|>0\big{\}}$$

$$({\mathcal{L}}^n└S,n)\mathrm{ap}D(f|S)(w)=DF(w)\text{for Ln almost all w∈S}$$

$$A$$

$$\displaystyle=\textstyle g\big{[}g^{-1}[B]\cap\{x\operatorname{:}\bigwedge_{m}(\operatorname{D}(F\circ g)(x)|\mathbf{R}^{m}\times\{0\})\neq 0\}\big{]}$$

$$R={\mathbf{R}}^m\cap \{\xi :(\xi ,\eta )\in g^{-1}[A]\}$$

$$\textstyle\mathscr{H}^{n-m}\big{(}f^{-1}[\{z\}]\operatorname{\sim}\bigcup_{i=1}^{\infty}K_{i}\big{)}=0\quad\text{for $\mathscr{H}^{m}$ almost all $z\in\mathbf{R}^{\nu}$}$$

$$Z={\mathbf{R}}^{\nu }\cap \{z:{\mathcal{H}}^{n-m}(f^{-1}[\{z\}])>0\},$$

$$\underset{y\to x}{\mathrm{limsup}}|y-x{|}^{-2}\mathrm{dist}(f(y)-f(x),U)<\infty \text{whenever x∈f−1[{z}]},$$

$$\underset{f[P]\ni \zeta \to z}{\mathrm{limsup}}|\zeta -z{|}^{-2}\mathrm{dist}(\zeta -z,U)<\infty .$$

$$c+t(v-c)\in V\text{whenever v∈V, c∈C, and 0<t≤1};$$

$$C\subset \{c+u:u\in \mathrm{Tan}(C,c)\}\subset \mathrm{aff}C;$$

$$\mathrm{Nor}(C,c)=\{v:{\mathbf{\xi }}_C(c+v)=c\}={\mathbf{R}}^n\cap \{v:\text{v∙(x−c)≤0 for x∈C}\}.$$

$$B=C\cap \{c:\text{c+v∈affC for some v∈Sn−1∩Nor(C,c)}\}.$$

$$\mathrm{dist}(v,\mathrm{aff}{C}_i)\to \mathrm{dist}(v,Q)\text{as i→∞ for v∈Rn},$$

$$D={\mathbf{R}}^n\cap \{d:\text{d∙c≤0 for c∈C}\},$$

$$\mathrm{dist}(d,U)\le -\gamma d\bullet v\text{for d∈D}.$$

$$\mathrm{dist}(b,U)\le -\gamma b\bullet v+(1+\gamma |v|)\mathrm{dist}(b,D)\text{for b∈Rn}.$$

$$\mathrm{Dis}(A)=({\mathbf{R}}^n\times {\mathbf{R}}^n)\cap \{(a,v):\text{a∈ClosA and ∣v∣=dist(a+v,A)}\}.$$

$$\big{\{}(a,|v|^{-1}v)\operatorname{:}\textup{$(a,v)\in\mathbf{R}^{n}\times\mathbf{R}^{n}$ and $0\neq v\in\operatorname{Dis}(A,a)$}\big{\}}$$

$$\text{either}v=0\text{or}q|v{|}^{-1}v\in \mathrm{Dis}(A,a),$$

$$|x-a|=\mathrm{dist}(x,A)\le r,|y-b|=\mathrm{dist}(y,A)\le r,$$

Full text

MSC-classes 2010.

52A20 (Primary); 28A78, 49Q15 (Secondary)

Keywords.

Second-order rectifiability, distance bundle, normal bundle, coarea formula, stratification.

1 Introduction

The main purpose of the present paper is to establish the following theorem; our notation is based on [Fed69, pp. 669–676], see the end of this introduction.

Structural theorem on the singularities of closed sets

$\mathrm{(see~{}\ref{thm:stratification-borel})}$.

Suppose $A$ is a closed subset of $\mathbf{R}^{n}$, for $a\in A$, $\operatorname{Dis}(A,a)$ is the set of $v\in\mathbf{R}^{n}$ satisfying $A\cap\{x\operatorname{:}|x-(a+v)|<|v|\}=\varnothing$, $m$ is an integer, $0\leq m\leq n$, and

[TABLE]

Then, $B$ can be $\mathscr{H}^{m}$ almost covered by the union of a countable collection of $m$ dimensional, twice continuously differentiable submanifolds of $\mathbf{R}^{n}$.

In the terminology of [San17, p. 2] for $m\geq 1$, the conclusion asserts that $B$ is countably $(\mathscr{H}^{m},m)$ rectifiable of class $2$. If $A$ is convex, then $B$ consists of the set of points, where the dimension of the normal cone of $A$ is at least $n-m$, see 4.14. Hence, our theorem contains Alberti’s structural theorem on the singularities of convex sets, see [Alb94, Theorem 3]. We also prove, that $B$ is a countably $m$ rectifiable Borel set, see 4.12; in particular, if $m\geq 1$, then $B$ can be covered (without exceptional set) by a countable family of images of Lipschitzian functions from $\mathbf{R}^{m}$ into $\mathbf{R}^{n}$, and, if $m=0$, then $B$ is countable.

Our approach rests on two pillars. The first may be stated as follows.

Parametric criterion for second-order rectifiability

$\mathrm{(see~{}\ref{corollary:rectifiability_criterion})}$.

Suppose $W$ is an $\mathscr{L}^{n}$ measurable subset of $\mathbf{R}^{n}$, $m$ is an integer, $1\leq m\leq n$, $f:W\to\mathbf{R}^{\nu}$ is a locally Lipschitzian map, $Z=\mathbf{R}^{\nu}\cap\{z\operatorname{:}\mathscr{H}^{n-m}(f^{-1}[\{z\}])>0\}$, and, for $\mathscr{H}^{m}$ almost all $z\in Z$, there exists an $m$ dimensional subspace $U$ of $\mathbf{R}^{\nu}$ satisfying

[TABLE]

Then, $Z$ can be $\mathscr{H}^{m}$ almost covered by the union of a countable collection of $m$ dimensional, twice continuously differentiable submanifolds of $\mathbf{R}^{\nu}$.

Notice that $f^{-1}[\{z\}]$ abbreviates $\{x\operatorname{:}f(x)=z\}$, see below. The key to reduce this criterion to the nonparametric case is the construction (in 2.1) of a countable collection $G$ of $m$ rectifiable subsets $P$ of $W$ with $\mathscr{H}^{m}(Z\operatorname{\sim}f[\bigcup G])=0$ such that, for each $P\in G$, the restriction $f|P$ is univalent and $(f|P)^{-1}$ is Lipschitzian. The nonparametric case was comprehensively studied in [San17]; however, for the present purpose, also [Sch09] would be sufficient (see 2.6).

The second pillar of the proof of the structural theorem is the next result that we state here for the special case of a convex set $A$. It concerns the relation of the nearest point projection, $\bm{\xi}_{A}$, with the tangent and normal cones of $A$.

A geometric observation for convex sets

$\mathrm{(see~{}\ref{lemma:relative_interior}\ with~{}\ref{lemma:convex}\,\eqref{item:convex:reach}\,\eqref{item:convex:normal},\ref{remark:dis_xi})}$.

If $A$ is a nonempty closed convex subset of $\mathbf{R}^{n}$, $m$ is an integer, $1\leq m<n$, $x\in\mathbf{R}^{n}\operatorname{\sim}A$, $a=\bm{\xi}_{A}(x)$, $\dim\operatorname{Nor}(A,a)\geq n-m$, $U$ is an $m$ dimensional subspace of $\mathbf{R}^{n}$, $U\subset\operatorname{Tan}(A,a)$, and $x-a$ belongs the relative interior of $\operatorname{Nor}(A,a)$, then

[TABLE]

This criterion and its generalisation to closed sets in 4.11 owe much to Federer’s treatment of sets of positive reach (a concept that embraces convex sets and submanifolds of class $2$) in [Fed59]. Since it is elementary, that the set $B$ in the structural theorem is countably $m$ rectifiable, the parametric criterion for second-order rectifiability then is readily applied with $f=\bm{\xi}_{A}|W$ for suitable $W$.

Connection to curvature measures

Instead of using second-order rectifiability properties, curvature properties can also be studied via general Steiner formulae. This approach was taken, for sets of positive reach and various more general classes of sets, by Federer in [Fed59], Stachó in [Sta79], Zähle in [Zäh86], Rataj and Zähle in [RZ01], and Hug, Last, and Weil in [HLW04]; in fact, [Sta79] and [HLW04] treat arbitrary closed subsets of Euclidean space. Accordingly, the natural question (under investigation by the second author) arises to characterise the relation of both notions of curvature.

Connection to varifold theory

The original motivation of the first author for the present study was to create a deeper understanding of a relation proven by Almgren in his area-mean-curvature characterisation of the sphere in [Alm86]. There, an equation relating the curvature measures (similar to those of [Zäh86]) of the convex hull of the support of a certain varifold to the perpendicular part of the mean curvature of the varifold is established in [Alm86, § 6 (2)]. The results of present paper shall serve as tools for further investigations of both authors of the second-order rectifiability properties of classes of varifolds.

Acknowledgements

The first author thanks the participants of an online reading seminar of [Alm86] for their early interest in these developments. The material of this paper originates from the PhD thesis of the second author, supervised by the first author, submitted at the University of Potsdam. The paper was written while both authors worked at the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) and the University of Potsdam.

Notation

Our notation and terminology is that of [Fed69, pp. 669–676], except that, as in [Kel75, p. 8], we denote the image of $A$ under a relation $r$ by

[TABLE]

2 Coarea formula

The purpose of the present section is to prove the parametric criterion for second-order rectifiability in 2.5. We begin by establishing a theorem that allows to construct univalent parametrisations from a Lipschitzian given one.

2.1 Theorem**.**

Suppose $W$ is an $\mathscr{L}^{n}$ measurable subset of $\mathbf{R}^{n}$, $m$ is an integer, $1\leq m\leq n$, and $f:W\rightarrow\mathbf{R}^{\nu}$ is a locally Lipschitzian map. Then, there exists a countable collection $G$ of compact subsets $P$ of $W$, such that $f|P$ is univalent and $(f|P)^{-1}$ is Lipschitzian, satisfying

[TABLE]

Moreover, each member of $G$ is contained in some $m$ dimensional affine plane.

Proof*.*

We firstly consider the special case that $W$ is a compact subset of $\mathbf{R}^{n}$. Choose $F:\mathbf{R}^{n}\to\mathbf{R}^{\nu}$ with $F|W=f$ and $\operatorname{Lip}F=\operatorname{Lip}f<\infty$ by Kirszbraun’s theorem [Fed69, 2.10.43]; in particular, $\operatorname{D}F$ is a Borel function whose domain is a Borel set by [Fed69, 3.1.2]. Defining $Z_{i}=\mathbf{R}^{\nu}\cap\{z\operatorname{:}\mathscr{H}^{n-m}(f^{-1}[\{z\}])\geq 1/i\}$ whenever $i$ is a positive integer, we note that

[TABLE]

Moreover, the sets $Z_{i}$ are Borel sets by [Fed69, 2.10.26] and $(\mathscr{H}^{m},m)$ rectifiable by [Fed69, 3.2.31]. We define, for every positive integer $i$, the class $\Omega_{i}$ to consist of all families $G$ of compact subsets $P$ of $f^{-1}[Z_{i}]$ such that

[TABLE]

whenever $P,Q\in G$, and such that

[TABLE]

whenever $P\in G$. Clearly, each member of $\Omega_{i}$ is countable. Using Hausdorff’s maximal principle (see [Kel75, p. 33]), we choose maximal elements $G_{i}$ of $\Omega_{i}$. The proof of the present case will be concluded by establishing

[TABLE]

For this purpose, fix such $i$ and define Borel sets $T=Z_{i}\operatorname{\sim}\bigcup\{f[P]\operatorname{:}P\in G_{i}\}$ and $S=f^{-1}[T]$. If $T$ had positive $\mathscr{H}^{m}$ measure, then, noting [Fed69, 2.10.35],

[TABLE]

would be a Borel set and have positive $\mathscr{L}^{n}$ measure by the coarea formula [Fed69, 3.2.22 (3)] with $W$, $Z$, and $f$ replaced by $S$, $T$, and $f|S$, since

[TABLE]

by [Fed69, 2.10.19 (4)].

Consequently, identifying $\mathbf{R}^{n}\simeq\mathbf{R}^{m}\times\mathbf{R}^{n-m}$, there would exist a linear isometry $g:\mathbf{R}^{n}\to\mathbf{R}^{n}$ such that $\mathscr{L}^{n}(A)>0$ with

[TABLE]

and, as $A$ would be a Borel set, $\eta\in\mathbf{R}^{n-m}$ so that $\mathscr{L}^{m}(R)>0$ with

[TABLE]

by Fubini’s theorem, see [Fed69, 2.6.2 (3)]. Since $R$ would be a Borel set, we could apply [Fed69, 3.2.2] to the function $h:\mathbf{R}^{m}\to\mathbf{R}^{\nu}$ defined by $h(\xi)=(F\circ g)(\xi,\eta)$ for $\xi\in\mathbf{R}^{m}$, and use the Borel regularity of $\mathscr{H}^{m}$ to construct a subset $P$ of $g[R\times\{\eta\}]$ with $G_{i}\cup\{P\}\in\Omega_{i}$, contrary to the maximality of $G_{i}$.

To treat the general case, we choose an increasing sequence of compact subsets $K_{i}$ of $\mathbf{R}^{n}$ with \mathscr{L}^{n}\big{(}W\operatorname{\sim}\bigcup_{i=1}^{\infty}K_{i}\big{)}=0. Since, in conjunction with [Fed69, 2.4.5], [Fed69, 2.10.25] applied with $A$ replaced by $W\operatorname{\sim}\bigcup_{i=1}^{\infty}K_{i}$ implies that

[TABLE]

and $\lim_{i\to\infty}\mathscr{H}^{n-m}(f^{-1}[\{z\}]\cap K_{i})=\mathscr{H}^{n-m}(f^{-1}[\{z\}])$ for such $z$, we readily infer the conclusion.

2.2 Remark*.*

The contradiction argument is inspired by [Fed69, 3.2.21].

2.3 Remark*.*

For the nearest point projection onto a set of positive reach, the idea of exhaustion by means of images from lower dimensional parts of the domain of $f$ is employed in [Fed59, 4.15 (3)]. The important additional feature of members $P$ in our collection $G$ is the Lipschitz continuity of $(f|P)^{-1}$.

2.4 Remark*.*

One readily verifies that 2.1 also holds with $m=0$, but this will not be needed in the present paper.

The parametric criterion for second-order rectifiability now reads as follows.

2.5 Corollary**.**

Under the hypotheses of 2.1, if

[TABLE]

and, for $\mathscr{H}^{m}$ almost all $z\in Z$, there exists an $m$ dimensional subspace $U$ of $\mathbf{R}^{\nu}$ satisfying

[TABLE]

then $Z$ can be $\mathscr{H}^{m}$ almost covered by the union of a countable collection of $m$ dimensional submanifolds of $\mathbf{R}^{\nu}$ of class $2$.

Proof*.*

Whenever $P\in G$, as $(f|P)^{-1}$ is Lipschitzian, we notice that, for $\mathscr{H}^{m}$ almost all $z\in Z\cap f[P]$, there exists an $m$ dimensional subspace $U$ of $\mathbf{R}^{\nu}$ such that

[TABLE]

Therefore, the conclusion follows from [San17, 5.3] and [Fed69, 3.1.15].

2.6 Remark*.*

With little additional effort, the final argument could have been based on [Sch09, A.1] instead of [San17, 5.3] and [Fed69, 3.1.15].

2.7 Remark*.*

In conjunction with the preceding corollary, the following observation will be useful. If $B$ is a countably $(\mathscr{H}^{m},m)$ rectifiable subset of $\mathbf{R}^{\nu}$, then, for $\mathscr{H}^{m}$ almost all $b\in B$, there exists an $m$ dimensional subspace $U$ of $\mathbf{R}^{\nu}$ such that $U\subset\operatorname{Tan}(B,b)$; in fact, [Fed69, 2.1.4, 3.1.21] reduce the problem to Borel sets $B$, in which case [Fed69, 2.10.19 (4), 3.2.17, 3.2.18] apply.

3 Convex sets

In the present section, we mainly collect some basic properties of convex sets and related definitions in 3.1–3.10 for convenient reference. Additionally, we note an observation concerning convex cones in 3.12–3.14.

3.1 Definition**.**

Suppose $A\subset\mathbf{R}^{n}$ and $x\in\mathbf{R}^{n}$. Then, the distance of $x$ to $A$ is denoted by $\operatorname{dist}(x,A)=\inf\{|x-a|\operatorname{:}a\in A\}$.

3.2 Remark*.*

If $A\neq\varnothing$, then $\operatorname{dist}(\cdot,A)$ is real valued and $\operatorname{Lip}\operatorname{dist}(\cdot,A)\leq 1$.

3.3 Remark*.*

If $R=(\mathbf{R}^{n}\times A)\cap\{(x,a)\operatorname{:}|x-a|=\operatorname{dist}(x,A)\}$, then, using 3.2, one verifies that \{a\operatorname{:}\textup{(x,a)\in R$for some$x\in B}\} is bounded whenever $B$ is a bounded subset of $\mathbf{R}^{n}$. Moreover, if $A$ is closed, so is $R$.

3.4 Definition** (see [Fed59, 4.1]).**

Suppose $A\subset\mathbf{R}^{n}$ and $U$ is the set of all $x\in\mathbf{R}^{n}$ such that there exists a unique $a\in A$ with $|x-a|=\operatorname{dist}(x,A)$. Then, the nearest point projection onto $A$ is the map $\bm{\xi}_{A}:U\to A$ characterised by the requirement $|x-\bm{\xi}_{A}(x)|=\operatorname{dist}(x,A)$ for $x\in U$.

3.5 Remark*.*

Using 3.3, we obtain that the function $\bm{\xi}_{A}$ is continuous. Moreover, if $A$ is closed, then $\operatorname{dmn}\bm{\xi}_{A}$ is a Borel set; in fact, one verifies, by means of 3.3, that the function mapping $x\in\mathbf{R}^{n}$ onto $d(x)=\operatorname{diam}\{a\operatorname{:}(x,a)\in R\}\in\overline{\mathbf{R}}$ is upper semicontinuous, and $\operatorname{dmn}\bm{\xi}_{A}=\{x\operatorname{:}d(x)=0\}$.

3.6 Definition** (see [Sch14, p. xix]).**

If $A\subset\mathbf{R}^{n}$, then $\operatorname{af{}f}A$ denotes the affine hull of $A$.

3.7 Definition** (see [Sch14, p. 7, p. xx]).**

Suppose $C$ is a convex subset of $\mathbf{R}^{n}$. Then, the dimension of $C$, denoted by $\dim C$, is defined to be the dimension of $\operatorname{af{}f}C$, and the relative boundary [interior] of $C$ is defined to be the boundary [interior] of $C$ relative to $\operatorname{af{}f}C$.

3.8 Remark*.*

If $V$ is the relative interior of $C$, then $V$ is convex, $\dim V=\dim C$, and

[TABLE]

in fact, reducing to the case $\operatorname{af{}f}C=\mathbf{R}^{n}$, this is [Sch14, 1.1.9, 1.1.10, 1.1.13].

3.9 Lemma**.**

Suppose $C$ is a nonempty closed convex subset of $\mathbf{R}^{n}$.

Then, the following four statements hold.

1. (1)

There holds $\operatorname{dmn}\bm{\xi}_{C}=\mathbf{R}^{n}$ and $\operatorname{Lip}\bm{\xi}_{C}\leq 1$. 2. (2)

If $c\in C$, then \operatorname{Tan}(C,c)=\mathbf{R}^{n}\cap\{u\operatorname{:}\textup{u\bullet v\leq 0$for$v\in\operatorname{Nor}(C,c)}\} and

[TABLE]

in particular, $\dim C=\dim\operatorname{Tan}(C,c)$. 3. (3)

If $c\in C$, then

[TABLE] 4. (4)

If $B$ is the relative boundary of $C$, then

[TABLE]

Proof*.*

(1) is asserted in [Fed69, 4.1.16]. In view of (1), the first equation and the first inclusion in (2) are contained in [Fed59, 4.8 (12)] and [Fed59, 4.18], respectively; the remaining items of (2) then follow. The first equation in (3) follows from (1) and [Fed59, 4.8 (12)]. The second equation in (3) follows from [KS80, I.2.3]. Finally, (4) is implied by [Sch14, 1.3.2].

3.10 Theorem**.**

Suppose $X=\mathbf{R}^{n}\cap\mathbf{B}(0,1)$, $F$ is the family of nonempty closed subsets of $X$ endowed with the Hausdorff metric, and G=F\cap\{C\operatorname{:}\textup{C is convex}\}.

Then, the following four statements hold.

1. (1)

The families $F$ and $G$ are compact. 2. (2)

The function mapping $(x,B)\in X\times F$ onto $\operatorname{dist}(x,B)\in\mathbf{R}$ is continuous. 3. (3)

The function mapping $C\in G$ onto $\dim C\in\mathbf{Z}$ is lower semicontinuous. 4. (4)

If \Phi=(G\times F)\cap\{(C,B)\operatorname{:}\textup{B$is the relative boundary of$C}\}, then $\Phi$ is a Borel function whose domain equals the Borel set $G\cap\{C\operatorname{:}\dim C\geq 1\}$.

Proof*.*

(1) is contained in [Fed69, 2.10.21]. (2) follows from 3.2. We observe that, in order to prove (3) and (4), it sufficient to establish the following assertion. If $k$ is an integer, $C_{i}$ is a sequence in $G$ with $\dim C_{i}=k$, $C\in G$, and $C_{i}\to C$ as $i\to\infty$, then $\dim C\leq k$ and, in case of equality with $k\geq 1$, also $\Phi(C)=\lim_{i\to\infty}\Phi(C_{i})$. For this purpose, we assume, possibly passing to a subsequence, that for some affine subspace $Q$ of $\mathbf{R}^{n}$

[TABLE]

and, if $k\geq 1$, that for some $B\in F$, we have $\Phi(C_{i})\to B$ as $i\to\infty$. It follows $C\subset Q$, whence we infer $\dim C\leq\dim Q\leq k$. Therefore, if $\dim C=k\geq 1$, then $Q=\operatorname{af{}f}C$ and we could assume, possibly replacing $C_{i}$ by $g_{i}^{-1}[C_{i}]$ for a sequence of isometries $g_{i}$ of $\mathbf{R}^{n}$ with $\lim_{i\to\infty}g_{i}(x)=x$ for $x\in\mathbf{R}^{n}$ and using 3.2, that $C_{i}\subset Q$ for each index $i$; in which case $\Phi(C)=B$ follows readily from 3.9 (4).

3.11 Remark*.*

We observe that (2)–(4) imply that, if $A$ is a Borel subset of $\mathbf{R}^{n}$ and $\Gamma:A\to G$ is a Borel function, then the set of $(a,v)\in A\times\mathbf{R}^{n}$ such that $v$ belongs to the relative interior of $\Gamma(a)$ is a Borel subset of $\mathbf{R}^{n}\times\mathbf{R}^{n}$.

The corollary to the next theorem on convex cones will be one of the ingredients to the geometric observation for convex sets described in the introduction.

3.12 Definition**.**

A subset $C$ of $\mathbf{R}^{n}$ is said to be a cone if and only if $\lambda c\in C$ whenever $0<\lambda<\infty$ and $c\in C$.

3.13 Theorem**.**

Suppose $C$ is a convex cone in $\mathbf{R}^{n}$,

[TABLE]

$U$* is an $m$ dimensional plane in $\mathbf{R}^{n}$, $U\subset D$, $\dim C\geq n-m$, and $v$ belongs to the relative interior of $C$.*

Then, $\dim C=n-m$ and there exists $0\leq\gamma<\infty$ satisfying

[TABLE]

Proof*.*

Defining V=\mathbf{R}^{n}\cap\{v\operatorname{:}\textup{u\bullet v=0$for$u\in U}\}, we see $C\subset V$ from [Fed59, 4.5], hence $\operatorname{af{}f}C=V$; in particular, $\dim C=n-m$. Since $D$ is closed under addition and $U\subset D$, $D$ is invariant under directions in $U$. Therefore, it is sufficient to prove the existence of $0\leq\gamma<\infty$ such that the inequality holds for $d\in D\cap V\cap\mathbf{S}^{n-1}$. If there were no such $\gamma$, then, by compactness, there would exist $d\in D\cap V\cap\mathbf{S}^{n-1}$ with $d\bullet v=0$ which would imply that $v$ belongs to the relative boundary of $C$, as $d\in\operatorname{af{}f}C$.

3.14 Corollary**.**

Under the hypotheses of 3.13, there holds

[TABLE]

Proof*.*

In view of 3.2 and 3.9 (1), one may apply 3.13 to $d=\bm{\xi}_{D}(b)$.

4 Distance bundle

In the present section, we introduce the distance bundle in 4.1–4.6; its nonzero directions correspond to the normal bundle employed by Hug, Last, and Weil in [HLW04], see 4.6. Then, we extend (in 4.7) some basic estimates from Federer’s treatment of sets of positive reach in [Fed59] which lead to an important one-sided estimate for the nearest point projection in 4.9. Finally, we derive the geometric observation, described for convex sets in the introduction, in 4.11, and the main structural theorem on the singularities of closed sets in 4.12.

4.1 Definition**.**

Suppose $A\subset\mathbf{R}^{n}$. Then, the distance bundle of $A$ is defined by

[TABLE]

Moreover, we let $\operatorname{Dis}(A,a)=\{v\operatorname{:}(a,v)\in\operatorname{Dis}(A)\}$ for $a\in\mathbf{R}^{n}$.

4.2 Remark*.*

Clearly, $\operatorname{Dis}(A)=\operatorname{Dis}(\operatorname{Clos}(A))$, $\operatorname{Dis}(A)$ is closed, and $0\in\operatorname{Dis}(A,a)$ if and only if $a\in\operatorname{Clos}A$. Moreover, $\operatorname{Dis}(A,a)$ is a convex subset of $\operatorname{Nor}(A,a)$ for $a\in\mathbf{R}^{n}$ by [Fed59, 4.8 (2)].

4.3 Remark*.*

If $X$ and $G$ are as in 3.10, then the function mapping $a\in\operatorname{Clos}A$ onto $\operatorname{Dis}(A,a)\cap X\in G$ is a Borel function; in fact, 4.2 implies that, in the terminology of [CV77, II.20], the function in question is an upper semicontinuous multifunction, whence the assertion follows by [CV77, III.3].

4.4 Remark*.*

If $a\in A$, $v\in\operatorname{Dis}(A,a)$, and $0\leq t<1$, then $\bm{\xi}_{A}(a+tv)=a$. In particular, $\bm{\xi}_{A}(a+v)=a$ whenever $v$ belongs to the relative interior of $\operatorname{Dis}(A,a)$, and $\operatorname{Dis}(A,a)$ is the closure of $\{v\operatorname{:}\bm{\xi}_{A}(a+v)=a\}$.

4.5 Remark*.*

In view of 4.4, we could have alternatively formulated our main theorem (see 4.12), for closed sets, in terms of the bundle $\{(a,v)\operatorname{:}\bm{\xi}_{A}(a+v)=a\}$ which would be more in line with Stachó’s definition of prenormals in [Sta79, p. 192]. Our choice of bundle is motivated by the fact that $\operatorname{Dis}(A)$ is closed.

4.6 Remark*.*

If $A$ is closed, then 4.4 yields that

[TABLE]

equals the normal bundle of $A$ defined in [HLW04, p. 239].

Basic estimates for the distance bundle are collected in the following theorem.

4.7 Theorem**.**

Suppose $A\subset\mathbf{R}^{n}$. Then, the following three statements hold.

1. (1)

If $0<q<\infty$, $a\in\operatorname{Clos}A$, $b\in\operatorname{Clos}A$, $v\in\mathbf{R}^{n}$, and

[TABLE]

then $(b-a)\bullet v\leq(2q)^{-1}|b-a|^{2}|v|$. 2. (2)

If $0<r<q<\infty$, $x\in\mathbf{R}^{n}$, $y\in\mathbf{R}^{n}$, $a\in A$, $b\in A$, and

[TABLE]

then $\bm{\xi}_{A}(x)=a$, $\bm{\xi}_{A}(y)=b$, and

[TABLE] 3. (3)

If $0<q<\infty$, $a\in\operatorname{Clos}A$, $b\in\operatorname{Clos}A$, $C$ is a convex cone in $\mathbf{R}^{n}$,

[TABLE]

and D=\mathbf{R}^{n}\cap\{u\operatorname{:}\textup{u\bullet v\leq 0$for$v\in C}\}, then

[TABLE]

Proof*.*

To prove (1), we assume $v\neq 0$, let $w=|v|^{-1}v$, and compute

[TABLE]

To prove (2), we notice that $a=\bm{\xi}_{A}(x)$ and $b=\bm{\xi}_{A}(y)$ by 4.4 and infer

[TABLE]

from applying (1) twice; once with $v$ replaced by $x-a$ and once with $a$, $b$, and $v$ replaced by $b$, $a$, and $y-b$. Therefore, we obtain

[TABLE]

whence we infer $|x-y|\geq|a-b|(q-r)/q$.

To prove (3), we suppose $a=0$. Whenever $v\in C$, we notice that

[TABLE]

by (1), and estimate

[TABLE]

Consequently, $\operatorname{dist}(b,C)^{2}\geq|b|^{2}-|b|^{4}/(4q^{2})$ and (3) is implied by [Fed59, 4.16].

4.8 Remark*.*

The proof is almost verbatim taken from [Fed59, 4.8 (7) (8), 4.18 (2)], where slightly stronger hypotheses were made.

Next, we derive a crucial one-sided estimate for the nearest point projection.

4.9 Corollary**.**

Suppose $A\subset\mathbf{R}^{n}$, $0<s<r<q<\infty$, and

[TABLE]

Then, there holds

[TABLE]

where $\kappa=(2s)^{-1}(1+2q/(q-r))^{2}$.

Proof*.*

We let $a=\bm{\xi}_{A}(x)$ and $b=\bm{\xi}_{A}(y)$, hence $a=x-|x-a|v$ and $b=y-|y-b|w$. Next, we estimate

[TABLE]

in case $\operatorname{dist}(x,A)=\operatorname{dist}(y,A)=s$; in fact, noting $\operatorname{dist}(y,A)\leq|y-a|$ and $|v|=|w|=1$, we obtain

[TABLE]

In the general case, we let (see 3.9 (1))

[TABLE]

notice $\alpha=a+sv$ and $\beta=b+sw$, and infer

[TABLE]

from 4.4, 3.9 (1), and 4.7 (2). Therefore, we may apply the previous case with $x$ and $y$ replaced by $\alpha$ and $\beta$ to deduce the conclusion.

4.10 Remark*.*

One could also derive a two-sided estimate; in fact, this is done in the submitted PhD thesis of the second author.

We now have all ingredients at our disposal to derive the geometric observation, formulated in the introduction for convex sets, in full generality.

4.11 Lemma**.**

Suppose $A\subset\mathbf{R}^{n}$, $0<q<\infty$, $m$ is an integer, $1\leq m<n$, $W$ is the set of $y\in\operatorname{dmn}\bm{\xi}_{A}$ satisfying

[TABLE]

$x\in W$, $a=\bm{\xi}_{A}(x)$, $\dim\operatorname{Dis}(A,a)\geq n-m$, $U$ is an $m$ dimensional subspace of $\mathbf{R}^{n}$, $U\subset\operatorname{Tan}(A,a)$, and

[TABLE]

Then,

[TABLE]

Proof*.*

Assume $a=0$, choose $s$ and $r$ such that $0<s<|x|<r<q$, and let $Q=\operatorname{af{}f}\operatorname{Dis}(A,0)$. Then, the set $X$ of all $v\in Q\operatorname{\sim}\{0\}$, such that $q|v|^{-1}v$ belongs to the relative interior of $\operatorname{Dis}(A,0)$, is relatively open in $Q$ and $x\in X$. This implies the existence of $\varepsilon>0$ such that the convex cone

[TABLE]

satisfies $C\cap\{v\operatorname{:}|v|=|x|\}\subset X$, hence

[TABLE]

in particular, $C\subset\operatorname{Nor}(A,0)$ by 4.2. We note that $\dim C=\dim Q\geq n-m$ and that $x$ belongs to the relative interior of $C$, as $Q\cap\mathbf{U}(x,\varepsilon)\subset C$. Abbreviating D=\mathbf{R}^{n}\cap\{d\operatorname{:}\textup{d\bullet c\leq 0$for$c\in C}\}, we observe $U\subset D$ from [Fed59, 4.5], and employing $0\leq\gamma<\infty$ from 3.13 with $v=x$, we estimate

[TABLE]

whenever $y\in W$ and $s\leq\operatorname{dist}(y,A)\leq r$ by 3.14, 4.9, 4.7 (3), and 4.7 (2), where $\kappa=(2s)^{-1}(1+2q/(q-r))^{2}$ and $\lambda=\gamma\kappa|x|+(1+\gamma|x|)2^{-1}q(q-r)^{-2}$. Finally, $x$ belongs to the interior of $W\cap\{y\operatorname{:}s\leq\operatorname{dist}(y,A)\leq r\}$ relative to $W$ by 3.2.

Finally, we establish the structural theorem on the singularities of closed sets; in fact, we may formulate it for arbitrary subsets of Euclidean space.

4.12 Theorem**.**

Suppose $A\subset\mathbf{R}^{n}$, $m$ is an integer, and $0\leq m\leq n$. Then,

[TABLE]

is a countably $m$ rectifiable Borel set which can be $\mathscr{H}^{m}$ almost covered by the union of a countable family of $m$ dimensional submanifolds of $\mathbf{R}^{n}$ of class $2$.

Proof*.*

Let $B=\{a\operatorname{:}\dim\operatorname{Dis}(A,a)\geq n-m\}$. We assume $A$ to be a nonempty closed set by 4.2, and also $m<n$. As $0\in\operatorname{Dis}(A,a)$ for $a\in A$ by 4.2, we obtain

[TABLE]

from 3.9 (2); in particular, $B$ is a Borel set by 3.10 (3) and 4.3. We define $N$ to be the set of all $(a,v)\in A\times\mathbf{R}^{n}$ such that $v$ belongs to the relative interior of $\operatorname{Dis}(A,a)\cap\mathbf{B}(0,1)$, hence $N$ is a Borel set by 3.11 and 4.3. By 4.4, we have

[TABLE]

Noting 3.2 and 3.5, we define $W_{i}$ to be the Borel set of all $x\in\bm{\xi}_{A}^{-1}[B]$ satisfying

[TABLE]

for every positive integer $i$. Then, $\bm{\xi}_{A}|W_{i}$ is locally Lipschitzian by 4.7 (2) and 3.2, and

[TABLE]

by 3.8. We observe that this implies that

[TABLE]

since $(\bm{\xi}_{A}|W_{i})^{-1}[\{\bm{\xi}_{A}(x)\}]$ is relatively open in $\{\bm{\xi}_{A}(x)+v\operatorname{:}v\in\operatorname{af{}f}\operatorname{Dis}(A,\bm{\xi}_{A}(x))\}$.

We choose a countable family $F$ of $m$ dimensional affine planes in $\mathbf{R}^{n}$ such that $Q\cap\bigcup F$ is dense in $Q$, whenever $Q$ is an affine subspace of $\mathbf{R}^{n}$ with $\dim Q\geq n-m$; in fact, one may take $F$ to be a countable dense subset in the family of all $m$ dimensional affine planes in $\mathbf{R}^{n}$. Thence, we deduce, employing 3.8, that

[TABLE]

in fact, whenever $a\in B$, we take $Q=\{a+v\operatorname{:}v\in\operatorname{af{}f}\operatorname{Dis}(A,a)\}$, pick a positive integer $i$ such that, for some $x\in Q$ with $0<|x-a|<i^{-1}$, we have that $i^{-1}|x-a|^{-1}(x-a)$ belongs to the relative interior of $\operatorname{Dis}(A,a)\cap\mathbf{B}(0,1)$, choose such $x$ within $\bigcup F$, and conclude $x\in W_{i}$ with $\bm{\xi}_{A}(x)=a$, as $(a,x-a)\in N$. It follows that $B$ is countably $m$ rectifiable.

To prove the remaining property of $B$, we assume $m\geq 1$. Then, in view of 2.7 and 4.11, we may apply 2.5 with $f=\bm{\xi}_{A}|W_{i}$ for every positive integer $i$ to obtain the conclusion.

4.13 Remark*.*

Our proof of the countable $m$ rectifiability follows [Fed59, 4.15 (3)], where the case of sets of positive reach was treated.

4.14 Remark*.*

If $A$ is a closed convex set, this property was proven, by different methods, in [Alb94, Theorem 3]; the agreement, in this case, of the normal bundle used there with our distance bundle follows from 3.9 (1) (3) and 4.4.

4.15 Remark*.*

For $1\leq m<n$, the preceding theorem may not be strengthened by replacing the distance bundle by the normal bundle, as is evident from considering a closed $m$ dimensional submanifold of $\mathbf{R}^{n}$ of class $1$ that meets every $m$ dimensional submanifold of $\mathbf{R}^{n}$ of class $2$ in a set of $\mathscr{H}^{m}$ measure zero; the existence of such $A$ follows from [Koh77].