Remarks on WDC sets

TL;DR

This paper investigates WDC sets in the plane, showing that their distance functions serve as DC auras, which leads to new insights about their structure and measurability within the space of compact sets.

Contribution

It proves that the distance function is a DC aura for WDC sets in \\mathbb{R}^2, establishing that locally WDC sets are WDC and that compact WDC sets form a Borel subset.

Findings

Distance function acts as a DC aura for WDC sets in \\mathbb{R}^2

Locally WDC sets in \\mathbb{R}^2 are WDC sets

Compact WDC sets form a Borel subset of all compact sets

Abstract

We study WDC sets, which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. Main results concern WDC sets $A\subset \mathbb{R}^2$. We prove that, for such $A$, the distance function $d_A= {\rm dist}(\cdot,A)$ is a `DC aura' for $A$, which implies that each locally WDC set in $\mathbb{R}^2$ is a WDC set. An another consequence is that compact WDC subsets of $\mathbb{R}^2$ form a Borel subset of the space of all compact sets.

Equations

$$d_A$$

$$f(x)=g(x)-a\Vert x{\Vert }^2(\text{resp.}f(x)=a\Vert x{\Vert }^2-g(x)),x\in C.$$

$$d_A$$

$$v\in {\partial }_{C}f(x)\text{ whenever}{x}_i\to x,v_i\in {\partial }_{C}f(x_i)\text{ and}{v}_i\to v.$$

$${\partial }_C(\alpha f)(x)=\alpha {\partial }_{C}f(x).$$

$$f^0(x,v):=\underset{y\to x,t\to 0+}{\mathrm{limsup}}\frac{f(y+tv)-f(y)}{t}$$

$$f^0(x,v)=\sup \{⟨v,\nu ⟩:\nu \in {\partial }_{C}f(x)\}.$$

$$\exists v\in S^{d-1},\rho >0\forall y\in U(x,\rho ),0<\alpha <\rho :\frac{f(y+\alpha v)-f(y)}{\alpha }\le -\epsilon .$$

$$\mathrm{dist}(0,{\partial }_{C}f(x))=-\sup \{⟨v,\nu ⟩:\nu \in {\partial }_{C}f(x)\}.$$

$$M\cap U(x,\rho )=x+C,$$

$$U(x,\rho )\setminus M=\bigcup _{i=1}^k\left(x+C_i\right).$$

$$|u|\le \frac{Lr}{\sqrt{1+L^2}}$$

$$L\ge \frac{f(t)-f(u)}{t-u}\ge \frac{g(t)-g(u)}{t-u}\text{for each}0<t<u,$$

$$v=g(u)\le -\sqrt{r^2-{\left(\frac{Lr}{\sqrt{1+L^2}}\right)}^2}=-\frac{r}{\sqrt{1+L^2}}.$$

$${\partial }_{C}d(0)=\mathrm{conv}\left\{\frac{1}{r}(-u,-v):(u,v)\in P\right\}\subset \mathbb{R}\times \left[\frac{1}{\sqrt{L^2+1}},\infty \right)$$

$$\tilde{f}(u):=\{\begin{array}{ll}f(u)& \text{if}0\le u\le \rho ,\\ f(\rho )& \text{if}\rho <u,\\ 2Ly& \text{if}u<0.\end{array}\text{and}\tilde{g}(u):=\{\begin{array}{ll}g(u)& \text{if}0\le u\le \rho ,\\ g(\rho )& \text{if}\rho <u,\\ -2Lu& \text{if}u<0.\end{array}$$

$$M_0:=\left\{(u,v)\in {\mathbb{R}}^2:u\ge 0,\tilde{g}(u)\le v\le \tilde{f}(u)\right\},$$

$$M_1:=\left\{(u,v)\in {\mathbb{R}}^2:u\ge 0,\tilde{f}(u)<v\right\}\cup \left\{(u,v)\in {\mathbb{R}}^2:u<0,-\frac{u}{2L}<v\right\},$$

$$M_2:=\left\{(u,v)\in {\mathbb{R}}^2:u\ge 0,\tilde{g}(u)>v\right\}\cup \left\{(u,v)\in {\mathbb{R}}^2:u<0,\frac{u}{2L}>v\right\}$$

$$M_3:=\left\{(u,v)\in {\mathbb{R}}^2:\frac{u}{L}<v<-\frac{u}{L}\right\}.$$

$$d_0(y)=0,d_1(y):=\mathrm{dist}(y,\mathrm{graph}\tilde{f}),d_2(y):=\mathrm{dist}(y,\mathrm{graph}\tilde{g}),d_3(y):=|y|.$$

$$\tilde{d}(y)=d_i(y)\text{ if}y\in M_i,0\le i\le 3,$$

$$d(y)=\{\begin{array}{ll}{\delta }_i(y)& \text{if}y\in C_i,\\ 0& \text{if}y\in M.\end{array}$$

$${\tilde{f}}_i(u):=\{\begin{array}{ll}{f}_i(u)& \text{if}u\in [-\rho ,\rho ],\\ f_i(-\rho )& \text{if}u<-\rho ,\\ f_i(\rho )& \text{if}u>\rho .\end{array}$$

$$d=2$$

$$F_{\sigma \delta \sigma }$$

$$K\in \mathcal{K}(B(0,1))$$

$$A:=\{f\in X:0\text{ is a weakly regular value of }f{|}_{U(0,4)}\},$$

$$D:=\{f\in X:f=g-h\text{for some convex Lipschitz functions}g,h\text{on}B(0,4)\}.$$

$$h(x):=\max (|x|-2,\alpha f(x)),x\in {\mathbb{R}}^d,\text{and}g:=f{\upharpoonright }_{B(0,4)}.$$

$$f(x):=\{\begin{array}{ll}\min (g(x),1),& \text{mbox}ifx\in U(0,4),\\ 1& \text{mbox}otherwise.\end{array}$$

Full text

The research was supported by GAČR 18-11058S

1. Introduction

In [10] (cf. also [8], [7] and [11]), the authors introduced the class of WDC sets which form a substantial generalization of sets with positive reach and still admit the definition of curvature measures. The following question naturally arises (see [8, Question 2, p. 829] and [7, 10.4.3]).

Question. Is the distance function $d_{A}={\rm dist}\,(\cdot,A)$ of each WDC set $A\subset{\mathbb{R}}^{d}$ a DC aura for $F$ (see Definition 2.3)?

We answer this question positively in the case $d=2$ (Theorem 3.3 below); it remains open for $d\geq 3$. The proof is based on a characterization (proved in [11]) of locally WDC sets in ${\mathbb{R}}^{2}$ and the main result of [12] which asserts that

[TABLE]

Recall that a function is called DC, if it is the difference of two convex functions and note that each set $A$ as in (1.1) is a WDC set.

Theorem 3.3 easily implies that each locally WDC set in ${\mathbb{R}}^{2}$ is WDC. Further, we use Theorem 3.3 to prove that compact WDC subsets of ${\mathbb{R}}^{2}$ form a Borel subset of the space of all compact sets of ${\mathbb{R}}^{2}$ (Theorem 4.1 (i)). The importance of this result is the fact that it suggests that (at least in ${\mathbb{R}}^{2}$) a theory of point processes on the space of compact WDC sets (analogous to the concept of point processes on the space of sets with positive reach introduced in [17]) can be build.

Concerning the compact WDC subsets of ${\mathbb{R}}^{d}$ for $d>2$, we are able to prove only a weaker fact that they form an analytic set (Theorem 4.1 (ii)) which is not probably sufficient for the above mentioned application.

2. Preliminaries

2.1. Basic definitions

The symbol $\mathbb{Q}$ denotes the set of all rational numbers. In any vector space $V$, we use the symbol [math] for the zero element. We denote by $B(x,r)$ ($U(x,r)$) the closed (open) ball with centre $x$ and radius $r$. The boundary and the interior of a set $M$ are denoted by $\partial M$ and $\mathrm{int}M$, respectively. A mapping is called $K$-Lipschitz if it is Lipschitz with a (not necessarily minimal) constant $K\geq 0$.

The metric space of all real-valued continuous functions on a compact $K$ (equipped with the usual supremum metric $\rho_{\sup}$) will be denoted $C(K)$.

In the Euclidean space ${\mathbb{R}}^{d}$, the norm is denoted by $|\cdot|$ and the scalar product by $\langle\cdot,\cdot\rangle$. By $S^{d-1}$ we denote the unit sphere in ${\mathbb{R}}^{d}$.

The distance function from a set $A\subset{\mathbb{R}}^{d}$ is $d_{A}\coloneqq{\rm dist}\,(\cdot,A)$ and the metric projection of $z\in{\mathbb{R}}^{d}$ to $A$ is $\Pi_{A}(z)\coloneqq\{a\in A:\,{\rm dist}\,(z,A)=|z-a|\}$.

2.2. DC functions

Let $f$ be a real function defined on an open convex set $C\subset{\mathbb{R}}^{d}$. Then we say that $f$ is a DC function, if it is the difference of two convex functions. Special DC functions are semiconvex and semiconcave functions. Namely, $f$ is a semiconvex (resp. semiconcave) function, if there exist $a>0$ and a convex function $g$ on $C$ such that

[TABLE]

We will use the following well-known properties of DC functions.

Lemma 2.1**.**

Let $C$ be an open convex subset of ${\mathbb{R}}^{d}$. Then the following assertions hold.

1. (i)

If $f:C\to{\mathbb{R}}$ and $g:C\to{\mathbb{R}}$ are DC, then (for each $a\in{\mathbb{R}}$, $b\in{\mathbb{R}}$) the functions $|f|$, $af+bg$, $\max(f,g)$ and $\min(f,g)$ are DC. 2. (ii)

Each locally DC function $f:C\to{\mathbb{R}}$ is DC. 3. (iii)

Each DC function $f:C\to{\mathbb{R}}$ is Lipschitz on each compact convex set $Z\subset C$. 4. (iv)

Let $f_{i}:C\to{\mathbb{R}}$, $i=1,\dots,m$, be DC functions. Let $f:C\to{\mathbb{R}}$ be a continuous function such that $f(x)\in\{f_{1}(x),\dots,f_{m}(x)\}$ for each $x\in C$. Then $f$ is DC on $C$.

Proof.

Property (i) follows easily from definitions, see e.g. [15, p. 84]. Property (ii) was proved in [9]. Property (iii) easily follows from the local Lipschitzness of convex functions. Assertion (iv) is a special case of [16, Lemma 4.8.] (“Mixing lemma”). ∎

It is well-known (cf. [12]) that if $\emptyset\neq A\subset{\mathbb{R}}^{d}$ is closed, then $d_{A}$ need not be DC; however (see, e.g., [2, Proposition 2.2.2]),

[TABLE]

2.3. Clarke generalized gradient

If $U\subset{\mathbb{R}}^{d}$ is an open set, $f:U\to{\mathbb{R}}$ is locally Lipschitz and $x\in U$, we denote by $\partial_{C}f(x)$ the generalized gradient of $f$ at $x$, which can be defined as the closed convex hull of all limits $\lim_{i\to\infty}f^{\prime}(x_{i})$ such that $x_{i}\to x$ and $f^{\prime}(x_{i})$ exists for all $i\in\mathbb{N}$ (see [3, S1.1.2]; $\partial_{C}f(x)$ is also called Clarke subdifferential of $f$ at $x$ in the literature). Since we identify $({\mathbb{R}}^{d})^{*}$ with ${\mathbb{R}}^{d}$ in the standard way, we sometimes consider $\partial_{C}f(x)$ as a subset of ${\mathbb{R}}^{d}$. We will repeatedly use the fact that the mapping $x\mapsto\partial_{C}f(x)$ is upper semicontinuous and, hence (see [3, Theorem 2.1.5]),

[TABLE]

We also use that $|u|\leq K$ whenever $u\in\partial_{C}f(x)$ and $f$ is $K$-Lipschitz on a neighbourhood of $x$. Obviously,

[TABLE]

Recall that

[TABLE]

and (see [3])

[TABLE]

We will need the following simple lemma.

Lemma 2.2**.**

Let $f$ be a Lipschitz function on an open set $G\subset{\mathbb{R}}^{d}$, $x\in G$ and $\varepsilon>0$.

1. (i)

If ${\rm dist}\,(0,\partial_{C}f(x))\geq 2\varepsilon$, then

[TABLE] 2. (ii)

If (2.6) holds, then ${\rm dist}\,(0,\partial_{C}f(x))\geq\varepsilon$.

Proof.

(i) Let ${\rm dist}\,(0,\partial_{C}f(x))\geq 2\varepsilon$. Since $\partial_{C}f(x)$ is convex, there exists (see e.g. [4, Theorem 1.5.]) $v\in S^{d-1}$ such that

[TABLE]

So, by (2.5), $f^{0}(x,v)\leq-2\varepsilon$ and thus (2.4) implies (2.6).

(ii) If (2.6) holds, choose corresponding $v\in S^{d-1}$ and $\rho>0$. Then $f^{0}(x,v)\leq-\varepsilon$ by (2.4). Consequently, by (2.5), $-|\nu|\leq\langle v,\nu\rangle\leq-\varepsilon$ for each $\nu\in\partial_{C}f(x)$ and so ${\rm dist}\,(0,\partial_{C}f(x))\geq\varepsilon$. ∎

2.4. WDC sets

WDC sets (see the definition below) which provide a natural generalization of sets with positive reach were defined in [10] using Fu’s notion of an “aura” of a set (see, e.g., [7] for more information). Note that the notion of a DC aura were defined in [10] and [8] by a formally different but equivalent way (cf. [11, Remark 2.12 (v)]).

Definition 2.3** (cf. [11], Definitions 2.8, 2.10).**

Let $U\subset{\mathbb{R}}^{d}$ be open and $f:U\to{\mathbb{R}}$ be locally Lipschitz. A number $c\in{\mathbb{R}}$ is called a weakly regular value of $f$ if whenever $x_{i}\to x$, $f(x_{i})>c=f(x)$ and $u_{i}\in\partial_{C}f(x_{i})$ for all $i\in\mathbb{N}$ then $\liminf_{i}|u_{i}|>0$.

A set $A\subset{\mathbb{R}}^{d}$ is called WDC if there exists a DC function $f:{\mathbb{R}}^{d}\to[0,\infty)$ such that $A=f^{-1}\{0\}$ and [math] is a weakly regular value of $f$. In such a case, we call $f$ a DC aura (for $A$).

A set $A\subset{\mathbb{R}}^{d}$ is called locally WDC if for any point $a\in A$ there exists a WDC set $A^{*}\subset{\mathbb{R}}^{d}$ that agrees with $A$ on an open neighbourhood of $a$.

(Note that a weakly regular value of $f$ need not be in the range of $f$, and so $\emptyset$ is clearly a WDC set by our definition.)

Note that a set $A\subset{\mathbb{R}}^{d}$ has a positive reach at each point if and only there exists a DC aura for $A$ which is even semiconvex ([1]).

3. Distance function of a WDC set in ${\mathbb{R}}^{2}$ is an aura

First we present (slightly formally rewritten) [11, Definition 7.9].

Definition 3.1**.**

1. (i)

A set $S\subset{\mathbb{R}}^{2}$ will be called a basic open DC sector (of radius $r$) if $S=U(0,r)\cap\{(u,v)\in{\mathbb{R}}^{2}:\ u\in(-\omega,\omega),v>f(u)\}$, where $0<r<\omega$ and $f$ is a DC function on $(-\omega,\omega)$ such that $f(0)=0$, $R(u)\coloneqq\sqrt{u^{2}+f^{2}(u)}$ is strictly increasing on $[0,\omega)$ and strictly decreasing on $(-\omega,0]$.

By an open DC sector (of radius $r$) we mean an image $\gamma(S)$ of a basic open DC sector $S$ (of radius $r$) under a rotation around the origin $\gamma$. 2. (ii)

A set of the form $\gamma(\{(u,v)\in{\mathbb{R}}^{2}:\ u\in[0,\omega),g(u)\leq v\leq f(u)\})\cap U(0,r)$, where $\gamma$ is a rotation around the origin, $0<r<\omega$ and $f,g:{\mathbb{R}}\to\mathbb{R}$ are DC functions such that $g\leq f$ on $[0,\omega)$, $f(0)=g(0)=f^{\prime}_{+}(0)=g^{\prime}_{+}(0)=0$ and the functions $R_{f}(u)\coloneqq\sqrt{u^{2}+f^{2}(u)}$, $R_{g}(u)\coloneqq\sqrt{u^{2}+g^{2}(u)}$ are strictly increasing on $[0,\omega)$, will be called a degenerated closed DC sector (of radius $r$).

We will use the following complete characterization of locally WDC sets in ${\mathbb{R}}^{2}$ ([11, Theorem 8.14]).

Theorem PRZ**.**

Let $M$ be a closed subset of $\mathbb{R}^{2}$. Then $M$ is a locally WDC set if and only if for each $x\in\partial M$ there is $\rho>0$ such that one of the following conditions holds:

1. (i)

$M\cap U(x,\rho)=\{x\}$, 2. (ii)

there is a degenerated closed DC sector $C$ of radius $\rho$ such that

[TABLE] 3. (iii)

there are pairwise disjoint open DC sectors $C_{1},\dots,C_{k}$ of radius $\rho$ such that

[TABLE]

Lemma 3.2**.**

Let $f$ be an $L$-Lipschitz function on $\mathbb{R}$. Denote $d\coloneqq{\rm dist}\,(\cdot,\operatorname{graph}f)$. Then $|\xi_{2}|\geq\frac{1}{\sqrt{L^{2}+1}}$ whenever $\xi=(\xi_{1},\xi_{2})\in\partial_{C}d(x)$ and $x\in\mathbb{R}^{2}\setminus\operatorname{graph}f$.

Proof.

Pick $x\in\mathbb{R}^{2}\setminus\operatorname{graph}f$. Without any loss of generality we can assume that $x=0$. We will assume that $f(0)<0$; the case $f(0)>0$ is quite analogous. Denote $r\coloneqq d(0)$ and $P\coloneqq\Pi_{\operatorname{graph}f}(0)$. Set $g(u)\coloneqq-\sqrt{r^{2}-u^{2}}$, $u\in[-r,r]$. Clearly $f\leq g$ on $[-r,r]$ and $(u,v)\in P$ if and only if $f(u)=g(u)=v$. We will show that

[TABLE]

To this end, suppose $(u,v)\in P$. If $u>0$, then

[TABLE]

and consequently $L\geq g^{\prime}_{-}(u)$. Therefore $u<r$ and $L\geq u(r^{2}-u^{2})^{-1/2}$. Analogously considering $g^{\prime}_{+}(u)$, we obtain for $u<0$ that $u>-r$ and $u(r^{2}-u^{2})^{-1/2}\geq-L$. In both cases we have $L\geq|u|(r^{2}-u^{2})^{-1/2}$ and an elementary computation gives (3.2).

Using (3.2) we obtain that if $(u,v)\in P$ then

[TABLE]

By [6, Lemma 4.2] and (3.3) we obtain

[TABLE]

and the assertion of the lemma follows. ∎

Theorem 3.3**.**

Let $M\neq\emptyset$ be a locally ${W\!DC}$ set in $\mathbb{R}^{2}$. Then the distance function $d_{M}$ is a DC aura for $M$. In particular, $M$ is a WDC set.

Proof.

Denote $d\coloneqq d_{M}$. For each $x\in\partial M$ choose $\rho=\rho(x)$ by Theorem PRZ. We will prove that

1. (a)

$d$ is DC on $U\left(x,\frac{\rho}{3}\right)$, 2. (b)

there is $\varepsilon=\varepsilon(x)>0$ such that $|\xi|\geq\varepsilon$ whenever $y\in U\left(0,\frac{\rho}{3}\right)\setminus M$ and $\xi\in\partial_{C}d(y)$.

Without any loss of generality we can assume that $x=0$.

If case (i) from Theorem PRZ holds, then $d(y)=|y|$, $y\in U\left(0,\frac{\rho}{3}\right)$, and so $d$ is convex and therefore DC on $U\left(0,\frac{\rho}{3}\right)$. Similarly, condition (b) holds as well, since if $y\in U\left(0,\frac{\rho}{3}\right)\setminus M$ and $\xi\in\partial_{C}d(y)$ then $\xi=\frac{y}{|y|}$ and so $|\xi|=1$.

If case (ii) from Theorem PRZ holds, we know that $M\cap U(0,\rho)$ is a degenerated closed DC sector $C$ of radius $\rho$. Let $\gamma$, $f$, $g$ and $\omega$ be as in Definition 3.1. Without any loss of generality we may assume that $\gamma$ is the identity map.

By Lemma 2.1 (iii) we can choose $L>0$ such that both $f$ and $g$ are $L$-Lipschitz on $[0,\rho]$ and define

[TABLE]

It is easy to see that both $\tilde{f}$ and $\tilde{g}$ are $2L$-Lipschitz and they are DC by Lemma 2.1 (iv).

Put

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

Clearly $\mathbb{R}^{2}=M_{0}\cup M_{1}\cup M_{2}\cup M_{3}$ and $M_{1}$, $M_{2}$, $M_{3}$ are open.

Set $\tilde{d}\coloneqq{\rm dist}\,(\cdot,M_{0})$ and, for each $y\in{\mathbb{R}}^{2}$, define

[TABLE]

Functions $d_{1}$ and $d_{2}$ are DC on $\mathbb{R}^{2}$ by (1.1), $d_{0}$ and $d_{3}$ are convex and therefore DC on $\mathbb{R}^{2}$.

Using (for $K=1/L,-1/L,1/(2L),-1/(2L)$) the facts that the lines with the slopes $K$ and $-1/K$ are orthogonal and $M_{0}\subset\{(u,v):u\geq 0,\;-Lu\leq v\leq Lu\}$, easy geometrical observations show that

[TABLE]

and so Lemma 2.1 (iv) implies that $\tilde{d}$ is DC.

Now pick an arbitrary $y\in\mathbb{R}^{2}\setminus M_{0}=M_{1}\cup M_{2}\cup M_{3}$ and choose $\xi=(\xi_{1},\xi_{2})\in\partial_{C}\tilde{d}(y)$. Using (3.4), we obtain that if $y\in M_{3}$ then $\xi=\frac{y}{|y|}$ and so $|\xi|=1$. Using Lemma 3.2, we obtain that if $y\in M_{1}\cup M_{2}$, then $|\xi|\geq|\xi_{1}|\geq\frac{1}{\sqrt{4L^{2}+1}}$.

Now, since $d=\tilde{d}$ on $U\left(0,\frac{\rho}{3}\right)$ both (a) and (b) follow.

It remains to prove (a) and (b) if case (iii) from Theorem PRZ holds. Let $C_{i}$, $i=1,\dots,k$, be the open DC sectors as in (iii). Denote $A_{i}\coloneqq\mathbb{R}^{2}\setminus C_{i}$ and define $\delta_{i}\coloneqq{\rm dist}\,(\cdot,A_{i})$, $i=1,\dots,k$.

Note that, for $y\in U\left(0,\frac{\rho}{3}\right)$, one has

[TABLE]

Therefore (by Lemma 2.1 (iv)) it is enough to prove that (a) and (b) hold with $d$ and $M$ being replaced by $\delta_{i}$ and $A_{i}$, respectively ($i=1,\dots,k$). Fix some $i\in\{1,\dots,k\}$. Without any loss of generality we can assume that $C_{i}$ is an basic open DC sector of radius $\rho$ with corresponding $f_{i}$ and $\omega_{i}$. Now define

[TABLE]

Then $\tilde{f}_{i}$ is Lipschitz and DC on $\mathbb{R}$. Put $\tilde{d}_{i}(y)={\rm dist}\,(y,\operatorname{graph}\tilde{f}_{i})$. Then $\tilde{d}_{i}$ is DC by (1.1) and [math] is a weakly regular value of $\tilde{d}_{i}$ by Lemma 3.2. And since $d_{i}=\tilde{d}_{i}$ on $U\left(0,\frac{\rho}{3}\right)$ we are done.

Since $d$ is locally DC on $\mathbb{R}^{2}\setminus M$ by (2.1) and on the interior of $M$ (trivially), (a) implies that $d$ is locally DC and so DC by Lemma 2.1 (ii). Further, (b) immediately implies that [math] is a weakly regular value of $d$ and thus $d=d_{M}$ is an aura for $M$. ∎

Remark 3.4**.**

By Theorem 3.3, in $\mathbb{R}^{2}$ locally ${W\!DC}$ sets and WDC sets coincide. This gives a partial answer to the part of [10, Problem 10.2] which asks whether the same is true in each ${\mathbb{R}}^{d}$.

4. Complexity of the system of WDC sets

In the following, we will work in each moment in an ${\mathbb{R}}^{d}$ with a fixed $d$, and so for simplicity we will use the notation, in which the dependence on $d$ is usually omitted.

The space of all nonempty compact subsets of ${\mathbb{R}}^{d}$ equipped with the usual Hausdorff metric $\rho_{H}$ is denoted by $\mathcal{K}$. It is well-known (see, e.g., [14, Proposition 2.4.15 and Corollary 2.4.16]) that $\mathcal{K}$ is a separable complete metric space. For a closed set $M\subset\mathbb{R}^{d}$, we set $\mathcal{K}(M)\coloneqq\{K\in\mathcal{K}:K\subset M\}$, which is clearly a closed subspace of $\mathcal{K}$. The set of all nonempty compact ${W\!DC}$ subsets of $M\subset\mathbb{R}^{d}$ will be denoted by ${W\!DC}(M)$.

In this section, we will prove the following theorem.

Theorem 4.1**.**

1. (i)

$WDC({\mathbb{R}}^{2})$* is an $F_{\sigma\delta\sigma}$ subset of $\mathcal{K}({\mathbb{R}}^{2})$.* 2. (ii)

$WDC({\mathbb{R}}^{d})$* is an analytic subset of $\mathcal{K}({\mathbb{R}}^{d})$ for each $d\in\mathbb{N}$.*

Before the proof of this theorem, we introduce some spaces, make a number of observations, and prove a technical lemma.

First observe that $WDC({\mathbb{R}}^{d})=\bigcup_{n=1}^{\infty}WDC(B(0,n))$ and so, to prove Theorem 4.1, it is sufficient to prove that, for each $r>0$,

[TABLE]

Further observe that it is sufficient to prove (LABEL:staci2) for $r=1$. Indeed, denoting $H(x)\coloneqq x/r,\ x\in{\mathbb{R}}^{d}$, it is obvious that $H^{*}:K\mapsto H(K)$ gives a homeomorphism of $\mathcal{K}(B(0,r))$ onto $\mathcal{K}(B(0,1))$ and $H^{*}(WDC(B(0,r))=WDC(B(0,1))$ (clearly $f$ is an aura for $K$ if and only if $f\circ H^{-1}$ is an aura for $H^{*}(K)$).

To prove (LABEL:staci2) for $r=1$, we will consider the space $X$ of all $1$-Lipschitz functions $f:B(0,4)\to[0,4]$ such that $f\geq 1$ on $B(0,4)\setminus U(0,3)$, equipped with the supremum metric $\rho_{\sup}$. Obviously, $X$ is a closed subspace of $C(B(0,4))$ and so it is a separable complete metric space.

The motivation for introducing $X$ is the fact that

[TABLE]

Since we are interested in $K\in WDC(B(0,1))$, we define also two subspaces of $X$:

[TABLE]

[TABLE]

Their complexity is closely related to the complexity of $WDC(B(0,1))$, as the following lemma indicate.

Lemma 4.2**.**

Let $\emptyset\neq K\subset B(0,1)\subset\mathbb{R}^{d}$ be compact. Then:

1. (i)

K is WDC if and only if there is a function $g\in D\cap A$ such that $K=g^{-1}(0)$. 2. (ii)

If $d=2$, then K is WDC if and only if $f_{K}\coloneqq d_{K}\restriction_{B(0,4)}\in D\cap A$.

Proof.

(i) Suppose first that $K$ is WDC and $f$ is an aura for $K$. Using Lemma 2.1(iii), we can chose $\alpha>0$ so small that the function $\alpha f$ is $1$-Lipschitz on $B(0,4)$ and $0\leq\alpha f(x)\leq 4$ for $x\in B(0,4)$. Set

[TABLE]

Then clearly $K=g^{-1}(0)$. Since $h$ is DC on ${\mathbb{R}}^{d}$ by Lemma 2.1(i), we obtain $g\in D$ by Lemma 2.1(iii). Finally, $g\in A$, since $g=\alpha f$ on $U(0,2)$.

Conversely, suppose that $K=g^{-1}(0)$ for some $g\in A\cap D$ and set

[TABLE]

Since $f$ is DC on $U(0,4)$ by Lemma 2.1 (i) and $f=1$ on ${\mathbb{R}}^{d}\setminus B(0,3)$, we see that $f$ is locally DC and so DC by Lemma 2.1 (ii). Since [math] is clearly a weakly regular value of $f$, we obtain that $f$ is an aura for $K$.

(ii) If $K$ is WDC, first note that $f_{K}\in X$ (see (4.2)). Since $d_{K}$ is an aura for $K$ by Theorem 3.3, we obtain immediately that $f_{K}\in A$, and also $f_{K}\in D$ by Lemma 2.1 (iii).

If $f_{K}\in A\cap D$, then $K$ is WDC by (i). ∎

For the application of Lemma 4.2(ii) we need the simple fact that

[TABLE]

Indeed, if $K_{1},K_{2}\in\mathcal{K}(B(0,1))$ with $\rho_{H}(K_{1},K_{2})<\varepsilon$ and $x\in B(0,4)$, then clearly $d_{K_{1}}(x)<d_{K_{2}}(x)+\varepsilon$, $d_{K_{2}}(x)<d_{K_{1}}(x)+\varepsilon$, and consequently $\rho_{\sup}(f_{K_{1}},f_{K_{2}})\leq\varepsilon$.

Further observe that

[TABLE]

To prove it, for each $n\in\mathbb{N}$ set

[TABLE]

Now observe that if $f\in D$ then we can choose $n\in\mathbb{N}$ and convex $n$-Lipschitz functions $g$, $h$ such that $f=g-h$, $g(0)=0$ and consequently $\|g\|\leq 4n$, $\|h\|\leq 4n+4$, and so $g,\,h\in C_{n}$. Consequently, $D=X\cap\bigcup_{n=1}^{\infty}(C_{n}-C_{n})$. Each $C_{n}$ is clearly closed in $C(B(0,4))$ and so it is compact in $C(B(0,4))$ by the Arzelà-Ascoli theorem. Consequently also $C_{n}-C_{n}=\sigma(C_{n}\times C_{n})$, where $\sigma$ is the continuous mapping $\sigma:(g,h)\mapsto g-h$, is compact, and (4.4) follows.

The most technical part of the proof of Theorem 4.1 is to show that $A$ is an $F_{\sigma\delta\sigma}$ subset of $X$. To prove it, we need some lemmas.

Lemma 4.3**.**

Let $f\in X$. Then $f\in A$ if and only

[TABLE]

Proof.

If (4.5) holds, then we easily obtain $f\in A$ directly from the definition of a weakly regular value.

To prove the opposite implication, suppose that $f\in A$ and (4.5) does not hold. Then there exist points $x_{n}\in f^{-1}(0,1/n),\ n\in\mathbb{N},$ and $\nu_{n}\in\partial_{C}f(x_{n})$ such that $|\nu_{n}|<1/n$. Choose a subsequence $x_{n_{k}}\to x\in B(0,4)$. Since $0\leq f(x_{n_{k}})<1/n_{k}$, we have $f(x_{n_{k}})\to f(x)=0$, and consequently $x\in U(0,4)$. Since $\nu_{n_{k}}\to 0$, we obtain that [math] is not a weakly regular value of $f|_{U(0,4)}$, which contradicts $f\in A$. ∎

Denote $\mathbb{Q}^{*}\coloneqq\mathbb{Q}\cap(0,1)$ and for every $\varepsilon\in\mathbb{Q}^{*}$ and $d\in\mathbb{N}$ pick a finite set $\mathfrak{S}^{d}_{\varepsilon}\subset S^{d-1}$ such that for every $v\in S^{d-1}$ there is some $\nu\in\mathfrak{S}^{d}_{\varepsilon}$ satisfying $|v-\nu|<\varepsilon$.

Lemma 4.4**.**

Let $f$ be a function from $X$. Then $f\in A$ if and only if

[TABLE]

Proof.

First suppose that (4.6) holds and choose $\varepsilon\in\mathbb{Q}^{*}$ by (4.6). We will show that

[TABLE]

To this end, consider an arbitrary $x\in f^{-1}(0,\varepsilon)$ and choose $p,q\in\mathbb{Q}^{*}$ such that $0<p<q<\varepsilon$ and $f(x)\in(p,q)$. Choose $\rho\in\mathbb{Q}^{*}$ which exists for $\varepsilon,p,q$ by (4.6). So there exists $\nu\in\mathfrak{S}^{d}_{\varepsilon}$ such that

[TABLE]

Therefore Lemma 2.2 (ii) gives that $|\nu|\geq\varepsilon$ for each $\nu\in\partial_{C}f(x)$. Thus (4.7) holds and so $f\in A$ by Lemma 4.3.

Now suppose $f\in A$. Using (4.5), we can choose $\varepsilon\in\mathbb{Q}^{*}$ such that

[TABLE]

To prove (4.6), consider arbitrary $p,q\in\mathbb{Q}^{*}$, $0<p<q<\varepsilon$. Using Lemma 2.2 (i), we easily obtain that for each $z\in K\coloneqq f^{-1}([p,q])$ there exist $\rho(z)>0$ and $v(z)\in S^{d-1}$ such that

[TABLE]

Choose $\rho\in\mathbb{Q}^{*}$ as a Lebesgue number (see [5]) of the open covering $\{U(z,\rho(z))\}_{z\in K}$ of the compact $K$. For an arbitrary $x\in U(0,4)$, either $f(x)\notin(p,q)$ or $x\in K$. In the second case, by the definition of Lebesgue number, there exists $z\in K$ such that $U(x,\rho)\subset U(z,\rho(z))$. Then clearly $\rho<\rho(z)$ and so (4.9) implies

[TABLE]

By the choice of $\mathfrak{S}_{\varepsilon}^{d}$ there is some $\nu\in\mathfrak{S}^{d}_{\varepsilon}$ such that $|v(z)-\nu|<\varepsilon$. By (4.10), for each $y\in U(x,\rho)$ and $0<\alpha<\rho$,

[TABLE]

Consequently, using $1$-Lipschitzness of $f\in X$, we obtain

[TABLE]

and so (4.6) holds. ∎

Corollary 4.5**.**

The set $A$ is an $F_{\sigma\delta\sigma}$ subset of $X$.

Proof.

For each quadruple $y\in\mathbb{R}^{d}$, $\nu\in S^{d-1}$, $\alpha>0$, $\varepsilon>0$ we set

[TABLE]

(Of course, we have $C(y,\nu,\alpha,\varepsilon)=\emptyset$ if $y\notin U(0,4)$ or $y+\alpha\nu\notin U(0,4)$.) Further, for each triple $x\in U(0,4)$, $0<p<q$, we set

[TABLE]

It is easy to see that both $C(y,\nu,\alpha,\varepsilon)$ and $D(x,p,q)$ are always closed subsets of $X$. It is easy to see that Lemma 4.4 is equivalent to

[TABLE]

Therefore, since $\mathbb{Q}^{*}$ is countable and each $\mathfrak{S}^{d}_{\varepsilon}$ is finite, we obtain that $A$ is an $F_{\sigma\delta\sigma}$ subset of $X$. ∎

The proof of Theorem 4.1. We know that it is sufficient to prove (LABEL:staci2) for $r=1$.

Suppose $d=2$. Then Lemma 4.2(ii) gives that $WDC(B(0,1))=\psi^{-1}(A\cap D)$, where $\psi:\mathcal{K}(B(0,4))\to X$ is the continuous mapping from (4.3). Since $A\cap D$ is an $F_{\sigma\delta\sigma}$ subset of $X$ by Corollary 4.5 and (4.4), we obtain (LABEL:staci2) for $r=1$ and $d=2$, and thus also assertion (i) of Theorem 4.1.

To prove assertion (ii) of Theorem 4.1, it is sufficient to prove that (in each ${\mathbb{R}}^{d}$) $WDC(B(0,1))$ is an analytic subset of $\mathcal{K}(B(0,1))$. To this end, consider the following subset $S$ of $\mathcal{K}(B(0,1))\times X$:

[TABLE]

By Lemma 4.2(i), $WDC(B(0,1))=\pi_{1}(S)$ (where $\pi_{1}(K,f)\coloneqq K$) and so it is sufficient to prove that $S$ is Borel. Denoting

[TABLE]

we have $S=Z\cap(\mathcal{K}(B(0,1))\times(A\cap D))$. So, since $A\cap D$ is Borel by Corollary 4.5 and (4.4), to prove that $S$ is Borel, it is sufficient to show that $Z$ is Borel in $\mathcal{K}(B(0,1))\times X$. To this end, denote for each $n\in\mathbb{N}$

[TABLE]

[TABLE]

Since clearly

[TABLE]

it is sufficient to prove that all $P_{n}$ and $Q_{n}$ are closed.

So suppose that $(K_{i},f_{i})\in\mathcal{K}(B(0,1))\times X,\ i=1,2,\dots,$ $(K,f)\in\mathcal{K}(B(0,1))\times X$, $\rho_{H}(K_{i},K)\to 0$ and $\rho_{\sup}(f_{i},f)\to 0$.

First suppose that $n\in\mathbb{N}$ and all $(K_{i},f_{i})\in P_{n}$. Choose $x_{i}\in K_{i}$ with $f_{i}(x_{i})\geq 1/n$. Choose a convergent subsequence $x_{i_{j}}\to x\in{\mathbb{R}}^{d}$. It is easy to see that $x\in K$. Since $|f_{i_{j}}(x_{i_{j}})-f(x_{i_{j}})|\to 0$ and $f(x_{i_{j}})\to f(x)$, we obtain $f_{i_{j}}(x_{i_{j}})\to f(x)$, and consequently $f(x)\geq 1/n$. Thus $(K,f)\in P_{n}$ and therefore $P_{n}$ is closed.

Second, suppose that $n\in\mathbb{N}$ and all $(K_{i},f_{i})\in P_{n}$. Choose $x_{i}\in B(0,4)$ such that ${\rm dist}\,(x_{i},K_{i})\geq 1/n$ and $f_{i}(x_{i})=0$. Choose a convergent subsequence $x_{i_{j}}\to x\in B(0,4)$. Since $|f_{i_{j}}(x_{i_{j}})-f(x_{i_{j}})|\to 0$ and $f(x_{i_{j}})\to f(x)$, we obtain $f(x)=0$. Now consider an arbitrary $y\in K$ and choose a sequence $y_{j}\in K_{i_{j}}$ with $y_{j}\to y$. Since $|x_{i_{j}}-y_{j}|\geq 1/n$ and $x_{i_{j}}\to x$, we obtain that $|y-x|\geq 1/n$ and consequently ${\rm dist}\,(x,K)\geq 1/n$. Thus $(K,f)\in Q_{n}$ and therefore $Q_{n}$ is closed.