On sets in ${\mathbb R}^d$ with DC distance function

TL;DR

This paper investigates the geometric properties of closed sets in Euclidean space whose distance functions are DC, establishing that in two dimensions, graphs of DC functions have DC distance functions, with partial results in higher dimensions.

Contribution

The paper proves that in two dimensions, graphs of DC functions have DC distance functions, and extends this to higher dimensions for semiconcave functions, highlighting open problems for general DC functions.

Findings

Graphs of DC functions in ${f R}^2$ have DC distance functions.

In higher dimensions, semiconcave functions also produce sets with DC distance functions.

The case for general DC functions in higher dimensions remains unresolved.

Abstract

We study closed sets $F \subset {\mathbb R}^d$ whose distance function $d_F:= {\rm dist}\,(\cdot,F)$ is DC (i.e., is the difference of two convex functions on ${\mathbb R}^d$). Our main result asserts that if $F \subset {\mathbb R}^2$ is a graph of a DC function $g:{\mathbb R}\to {\mathbb R}$, then $F$ has the above property. If $d>1$, the same holds if $g:{\mathbb R}^{d-1}\to {\mathbb R}$ is semiconcave, however the case of a general DC function $g$ remains open.

Equations

$$(d_F{)}^2$$

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

$$g_{+}^{\prime }(x,v)+g_{+}^{\prime }(x,-v)\le 0,x\in U,v\in {\mathbb{R}}^d,\text{if}g\text{is locally semiconcave on}U.$$

$$d_A$$

$${\gamma }_{+}^{\prime }(x,v)+{\gamma }_{+}^{\prime }(x,-v)\le 0,x\in U,v\in {\mathbb{R}}^d,$$

$$\begin{gathered} \text{ graphγ is covered by graphs of a finite number} \\ \text{of concave functions defined on U.} \end{gathered}$$

$$h_{-}^{\prime }(x)\le h_{+}^{\prime }(x),x\in (a,b).$$

$$\begin{gathered} \forall x_0\in (a,b)\exists \delta >0\forall x:(x\in (x_0,x_0+\delta )\Rightarrow h_{+}^{\prime }(x)\ge h_{+}^{\prime }(x_0)) \\ \wedge (x\in (x_0-\delta ,x_0)\Rightarrow h_{+}^{\prime }(x)\le h_{+}^{\prime }(x_0)). \end{gathered}$$

$$x_n<x_0\text{and}{h}_{+}^{\prime }(x_n)>h_{+}^{\prime }(x_0)\text{for each}n\in \mathbb{N}$$

$$x_n>x_0\text{and}{h}_{+}^{\prime }(x_n)<h_{+}^{\prime }(x_0)\text{for each}n\in \mathbb{N}.$$

$$h_{+}^{\prime }(x_n)=(h_{\alpha }{)}_{+}^{\prime }(x_n)\le (h_{\alpha }{)}_{-}^{\prime }(x_0)=h_{-}^{\prime }(x_0)\le h_{+}^{\prime }(x_0),$$

$$V=\{(x,y):x\ge 0,|y|\le x\tan (\alpha /2)\}.$$

$$\phi (x,y):=\sqrt{x^2+y^2}-x,(x,y)\in V.$$

$$\phi \text{is Lipschitz with constant}\sqrt{2}\tan (\alpha /2).$$

$$\left|\frac{\partial \phi }{\partial x}(x,y)\right|=\left|\frac{x}{\sqrt{x^2+y^2}}-1\right|=\frac{y^2}{(x+\sqrt{x^2+y^2})\sqrt{x^2+y^2}}\le \frac{|y|}{x}\le \tan (\alpha /2),$$

$$\left|\frac{\partial \phi }{\partial y}(x,y)\right|=\frac{|y|}{\sqrt{x^2+y^2}}\le \frac{|y|}{x}\le \tan (\alpha /2).$$

$$d$$

$$n_0\ge 6\text{and}|f_n(0)-f(0)|<\frac{1}{10}\text{for each}n\ge n_0.$$

$$\text{each}{c}_n\text{is Lipschitz with constant}{L}^{\ast }\text{and}$$

$$c_n^{\ast }:=d_n+c_n\text{is concave on}\overline{U}.$$

$$s_i:=\frac{f_n(x_{i+1})-f_n(x_i)}{x_{i+1}-x_i}\text{and}{\beta }_i:=\arctan s_i,i=0,\dots ,n-1.$$

$$\sum _{i=1}^{n-1}|s_i-s_{i-1}|\le 4L.$$

$${\tilde{s}}_i:=\frac{g_n(x_{i+1})-h_n(x_i)}{x_{i+1}-x_i},s_i^{\ast }:=\frac{h_n(x_{i+1})-h_n(x_i)}{x_{i+1}-x_i},i=0,\dots ,n-1,$$

$$\sum _{i=1}^{n-1}|{\tilde{s}}_i-{\tilde{s}}_{i-1}|={\tilde{s}}_n-{\tilde{s}}_1\le 2L\text{and}\sum _{i=1}^{n-1}|s_i^{\ast }-s_{i-1}^{\ast }|=s_n^{\ast }-s_1^{\ast }\le 2L.$$

$${\alpha }_i=|{\beta }_i-{\beta }_{i-1}|\le |\tan ({\beta }_i)-\tan ({\beta }_{i-1})|=|s_i-s_{i-1}|,$$

$$\sum _{i=1}^{n-1}{\alpha }_i\le 4L.$$

$$\tan \left(\frac{{\alpha }_i}{2}\right)\le \frac{{\alpha }_i}{2}\cdot \frac{L}{\arctan L}.$$

$$\sum _{i=1}^{n-1}\sqrt{2}\tan \left(\frac{{\alpha }_i}{2}\right)\le \frac{2\sqrt{2}{L}^2}{\arctan L}=:M.$$

$$(d_n{)}_{+}^{\prime }(x,v)\in \mathbb{R}\text{exists for every}x,v\in {\mathbb{R}}^2.$$

$$V_i:=\{z\in {\mathbb{R}}^2:⟨z-z_i,z_{i+1}-z_i⟩\le 0,⟨z-z_i,z_{i-1}-z_i⟩\le 0\},$$

Full text

The research was supported by GAČR 18-11058S

1. Introduction

Let $F\neq\varnothing$ be a closed subset of ${\mathbb{R}}^{d}$ and let $d_{F}\coloneqq{\rm dist}\,(\cdot,F)$ be its distance function. Recall that a function on ${\mathbb{R}}^{d}$ is called DC, if it is the difference of two convex functions. It is well-known (see, e.g., [1, p. 976]) that

[TABLE]

However, the distance function of some interesting special $F\subset{\mathbb{R}}^{d}$ is DC; it is true for example for $F$ from Federer’s class of sets with positive reach, see (4.5).

Our article was motivated by [1] and by the following question which naturally arises in the theory of WDC sets (see [8, Question 2, p. 829] and [7, 10.4.3]).

Question.  Is $d_{F}$ a DC function if $F$ is a graph of a $DC$ function $g:{\mathbb{R}}^{d-1}\to{\mathbb{R}}$?

Note that WDC sets form a substantial generalization of sets with positive reach and still admit the definition of curvature measures (see [11] or [7]) and $F$ as in Question is a natural example of a WDC set in ${\mathbb{R}}^{d}$.

Our main result (Theorem 3.3) gives the affirmative answer to Question in the case $d=2$; the case $d>2$ remains open. However, known results relatively easily imply that the answer is positive if $g$ in Question is semiconcave (Corollary 4.5).

In [13] we show that our main result has some interesting consequences for WDC subsets of ${\mathbb{R}}^{2}$, in particular that these sets have DC distance functions.

In Section 2 we recall some notation and needed facts about DC functions. In Section 3 we prove our main result (Theorem 3.3). In last Section 4, we prove a number of further results on the system of sets in ${\mathbb{R}}^{d}$ which have DC distance function, including Corollary 4.5 mentioned above.

We were not able to prove a satisfactory complete characterisation of sets $F\subset\mathbb{R}^{2}$ with DC distance function, but we believe that our methods and results should lead to such a characterisation. However, in our opinion, the case of $F\subset\mathbb{R}^{d}$, $d\geq 3$, needs some new ideas.

2. Preliminaries

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 $K\geq 0$.

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 ${\mathbb{R}}^{d}$.

If $x,y\in{\mathbb{R}}^{d}$, the symbol $[x,y]$ denotes the closed segment (possibly degenerate). If also $x\neq y$, then $l(x,y)$ denotes the line joining $x$ and $y$.

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 $\Pi_{A}(z)\coloneqq\{a\in A:\,{\rm dist}\,(z,A)=|z-a|\}$.

If $f$ is defined in ${\mathbb{R}}^{d}$, we use the notation $f^{\prime}_{+}(x,v)$ for the one sided directional derivative of $f$ at $x$ in direction $v$.

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 $C$. 5. (v)

Each $\mathcal{C}^{2}$ function $f:C\to\mathbb{R}$ is DC.

Proof.

Property (i) follows easily from definitions, see e.g. [17, 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 [18, Lemma 4.8.] (“Mixing lemma”). To prove (v) observe that (e.g. by [2, Prposition 1.1.3 (d)]) each $C^{2}$ function is locally semiconcave and therefore locally DC, hence, DC by (ii). ∎

By well-known properties of convex and concave functions, we easily obtain that each locally DC function $f$ on an open set $U\subset{\mathbb{R}}^{d}$ has all one-sided directional derivatives finite and

[TABLE]

Recall that if $\varnothing\neq A\subset{\mathbb{R}}^{d}$ is closed, then $d_{A}$ need not be DC; however (see, e.g., [2, Proposition 2.2.2]),

[TABLE]

3. Main result

In the proof of Theorem 3.3 below we will use the following simple “concave mixing lemma”.

Lemma 3.1**.**

Let $U\subset{\mathbb{R}}^{d}$ be an open convex set and let $\gamma:U\to{\mathbb{R}}$ have finite one-sided directional derivatives $\gamma_{+}^{\prime}(x,v)$, ($x\in U,\ v\in{\mathbb{R}}^{d}$). Suppose that

[TABLE]

and that

[TABLE]

Then $\gamma$ is a concave function.

Proof.

Since $\gamma$ is clearly concave if each function $t\mapsto\gamma(a+tv),\ (a\in C,v\in S^{d-1})$ is concave on its domain, it is sufficient to prove the case $d=1$, $C=(a,b)$. Set $h(x)\coloneqq-\gamma(x),\ x\in(a,b)$; we need to prove that $h$ is convex. Observe that (3.1) easily implies the condition

[TABLE]

and (3.2) implies that there exists a finite set $\{h_{\alpha}:\ \alpha\in A\}$ of convex functions on $(a,b)$ such that $\operatorname{graph}h\subset\bigcup\{\operatorname{graph}h_{\alpha}:\ \alpha\in A\}$. To prove the convexity of $h$, it is sufficient to show that the function $h^{\prime}_{+}$ is nondecreasing on $(a,b)$ (see e.g. [16, Chap. 5, Prop. 18, p. 114]); equivalently (it follows e.g. from [10, Chap. IX, §7, Lemma 1, p. 266]) to prove that

[TABLE]

So suppose, to the contrary, that (3.4) does not hold; then there exists a sequence $x_{n}\to x_{0}$ such that either

[TABLE]

or

[TABLE]

Since $h$ is clearly continuous, each set $F_{\alpha}\coloneqq\{x\in(a,b):\ h_{\alpha}(x)=h(x)\}$, $\alpha\in A$, is closed in $(a,b)$. Since $A$ is finite, it is easy to see that for each $n\in\mathbb{N}$ there exists $\alpha(n)\in A$ such that $x_{n}\in F_{\alpha(n)}$ and $x_{n}$ is a right accumulation point of $F_{\alpha(n)}$. Using finiteness of $A$ again, we can suppose that there exists $\alpha\in A$ such that $\alpha(n)=\alpha$, $n\in\mathbb{N}$ (otherwise we could consider a subsequence of $(x_{n})$).

Now suppose that (3.5) holds. Since $x_{n}\in F_{\alpha},\ n=0,1,\dots,$ we obtain that $h^{\prime}_{+}(x_{n})=(h_{\alpha})_{+}^{\prime}(x_{n}),\ n\in\mathbb{N}$, and $h^{\prime}_{-}(x_{0})=(h_{\alpha})_{-}^{\prime}(x_{0})$. Using also the convexity of $h_{\alpha}$ and (3.3), we obtain

[TABLE]

which contradicts (3.5). Since the case when (3.6) holds is quite analogous, neither (3.5) nor (3.6) is possible and so we are done. ∎

We will need also the following easy lemma.

Lemma 3.2**.**

Let $V$ be a closed angle in ${\mathbb{R}}^{2}$ with vertex $v$ and measure $0<\alpha<\pi$. Then there exist an affine function $A$ on ${\mathbb{R}}^{2}$ and a concave function $\psi$ on ${\mathbb{R}}^{2}$ which is Lipschitz with constant $\sqrt{2}\tan(\alpha/2)$ such that $|z-v|+\psi(z)=A(z),\ z\in V$.

Proof.

We can suppose without any loss of generality that $v=(0,0)$ and

[TABLE]

Then $|z-v|=\sqrt{x^{2}+y^{2}}$ for $z=(x,y)$. Define the convex function

[TABLE]

We will show that

[TABLE]

To this end estimate, for $(x,y)\in\mathrm{int}\,V$,

[TABLE]

[TABLE]

Thus $|\operatorname{grad}\varphi(x,y)|\leq\sqrt{2}\tan(\alpha/2)$ for $(x,y)\in\mathrm{int}\,V$ and (3.7) follows. So $\varphi$ has a convex extension $\tilde{\varphi}$ to ${\mathbb{R}}^{2}$ which is also Lipschitz with constant $\sqrt{2}\tan(\alpha/2)$ (see, e.g., [4, Theorem 1]). Now we can put $\psi\coloneqq-\tilde{\varphi}$, since $\sqrt{x^{2}+y^{2}}+\psi(x,y)=x=:A(x,y),\ (x,y)\in V$. ∎

Theorem 3.3**.**

Let $f:{\mathbb{R}}\to{\mathbb{R}}$ be a DC function. Then the distance function $d\coloneqq{\rm dist}\,(\cdot,\operatorname{graph}f)$ is DC on ${\mathbb{R}}^{2}$.

Proof.

By (2.2), $d$ is locally DC on ${\mathbb{R}}^{2}\setminus\operatorname{graph}f$. So, by Lemma 2.1 (ii), it is sufficient to prove that, for each $z\in\operatorname{graph}f$, the distance function $d$ is DC on a convex neighbourhood of $z$. Since we can clearly suppose that $z=(0,f(0))$, it is sufficient to prove that

[TABLE]

Write $f=g-h$, where $g$, $h$ are convex functions on ${\mathbb{R}}$. For each $n\in\mathbb{N}$, consider the equidistant partition $D_{n}=\{x^{n}_{0}=-1<x^{n}_{1}<\dots<x^{n}_{n}=1\}$ of $[-1,1]$. Let $g_{n}$, $h_{n}$ be the piece-wise linear function on $[-1,1]$ such that $g_{n}(x^{n}_{i})=g(x^{n}_{i})$, $h_{n}(x^{n}_{i})=h(x^{n}_{i})$ ($0\leq i\leq n$) and $g_{n}$, $h_{n}$ are affine on each interval $[x_{i-1},x_{i}]$ ($1\leq i\leq n$). Put $f_{n}\coloneqq g_{n}-h_{n}$ and $d_{n}\coloneqq{\rm dist}\,(\cdot,\operatorname{graph}f_{n})$. Choose $L>0$ such that both $g\restriction_{[-1,1]}$ and $h\restriction_{[-1,1]}$ are $(L/2)$-Lipschitz and observe that all $g_{n}$, $h_{n}$, $f_{n}$ are $L$-Lipschitz. Since $f_{n}$ uniformly converge to $f$ on $[-1,1]$, we easily see that $d_{n}\to d$ on $\overline{U}$.

Choose an integer $n_{0}$ such that

[TABLE]

We will prove that there exist $L^{*}>0$ and concave functions $c_{n}$ ($n\geq n_{0}$) on $\overline{U}$ such that

[TABLE]

[TABLE]

Then we will done, since (3.10) and (3.11) easily imply (3.8). Indeed, we can suppose that $c_{n}((0,f(0)))=0$ and, using Arzelà-Ascoli theorem, we obtain that there exists an increasing sequence of indices $(n_{k})$ such that $c_{n_{k}}\to c$, where $c$ is a continuous concave function on $\overline{U}$. So $d_{n_{k}}+c_{n_{k}}\to d+c=:c^{*}$ on $\overline{U}$. Using (3.11), we obtain that $c^{*}$ is concave and thus $d=c^{*}-c$ is DC on $U$.

To prove the existence of $L^{*}$ and $(c_{n})$, fix an arbitrary $n\geq n_{0}$. For brevity denote $\Pi\coloneqq\Pi_{\operatorname{graph}f_{n}}$ and put $x_{i}\coloneqq x^{n}_{i}$, $z_{i}\coloneqq(x_{i},f_{n}(x_{i})),\ i=0,\dots,n$. For $i=1,\dots,n-1$, let $0\leq\alpha_{i}<\pi$ be the angle between the vectors $z_{i}-z_{i-1}$ and $z_{i+1}-z_{i}$. Denote

[TABLE]

Then clearly $\alpha_{i}=|\beta_{i}-\beta_{i-1}|$. One of the main ingredients of the present proof is the easy fact that

[TABLE]

It immediately follows from the well-known estimate of (the “convexity”) $K_{a}^{b}(f_{n})$ (see [15, p. 24, line 5]). To give, for completeness, a direct proof, denote

[TABLE]

and observe that the finite sequences $(\tilde{s}_{i})$, $(s^{*}_{i})$ are nondecreasing. Consequently

[TABLE]

Since $s_{i}=\tilde{s}_{i}-\tilde{s}^{*}_{i}$, (3.12) easily follows.

Since

[TABLE]

we obtain

[TABLE]

Since $|\beta_{i}|\leq\arctan L$, we have $\alpha_{i}/2\leq\arctan L$. Further, since the function $\tan$ is convex on $[0,\pi/2)$, the function $s(x)=\tan x/x$ is increasing on $(0,\pi/2)$. These facts easily imply

[TABLE]

Thus we obtain by (3.13)

[TABLE]

Further observe that each $d_{n}$ is DC on ${\mathbb{R}}^{2}$ and consequently

[TABLE]

Indeed, since each segment $[z_{i-1},z_{i}]$ is a convex set, by the well known fact the distance functions ${\rm dist}\,(\cdot,[z_{i-1},z_{i}])$, $i=1,\dots,n,$ are convex and consequently $d_{n}$ is DC by (4.3) below.

If $\alpha_{i}\neq 0$, set

[TABLE]

which is clearly a closed angle with vertex $z_{i}$ and measure $\alpha_{i}$. Let $\psi_{i}$ and $A_{i}$ be the (concave and affine) functions on ${\mathbb{R}}^{2}$ which correspond to $V_{i}$ by Lemma 3.2. If $\alpha_{i}=0$, put $\psi_{i}(z)\coloneqq 0,\ z\in{\mathbb{R}}^{2}$.

Now set

[TABLE]

Then $\eta_{n}$ is a concave function on ${\mathbb{R}}^{2}$ and Lemma 3.2 with (3.14) imply that

[TABLE]

and, if $\alpha_{i}\neq 0$,

[TABLE]

The concave function $c_{n}$ with properties (3.10), (3.11) will be defined as $c_{n}(x)\coloneqq\eta_{n}(x)+\xi_{n}(x),\ x\in\overline{U}$, where the concave function $\xi_{n}$ on $A\coloneqq(-1,1)\times{\mathbb{R}}$ will be defined to “compensate the non-concave behaviour of $d_{n}$ at points of $\operatorname{graph}f_{n}$” in the sense that, for each point $z\in A\cap\operatorname{graph}f_{n}$,

[TABLE]

We set, for $(x,y)\in A$,

[TABLE]

Obviously,

[TABLE]

Further, for $(x,y)\in A$,

[TABLE]

which shows that $p_{n}$ is a DC function and $p_{n}+\xi_{n}$ is concave. Consequently, for each $z\in A$ and $v\in{\mathbb{R}}^{2}$,

[TABLE]

Since, for each point $z\in\operatorname{graph}f_{n}\cap A$, we have $d_{n}(z)=p_{n}(z)=0$ and for each $(x,y)\in A$ clearly $d_{n}(x,y)\leq|(x,y)-(x,f_{n}(x))|=p_{n}(x,y)$, we easily obtain (for each $v\in{\mathbb{R}}^{2}$)

[TABLE]

which, together with (3.20), implies (3.18).

Now set

[TABLE]

By (3.16) and (3.19) we obtain that (3.10) holds with $L^{*}\coloneqq M+2L+1$.

To prove (3.11), it is clearly sufficient to show that $\gamma=c_{n}^{*}:=d_{n}+c_{n}$ is concave on $U$; we will prove it by Lemma 3.1.

First we verify the validity of (3.1) for each $z\in U$. If $z\notin\operatorname{graph}f_{n}$, then (3.1) holds by (2.1), since $\gamma=d_{n}+\eta_{n}+\xi_{n}$ on $U$, $d_{n}$ is locally semiconcave on ${\mathbb{R}}^{2}\setminus\operatorname{graph}f_{n}$ and $\eta_{n}+\xi_{n}$ is concave on $U$. If $z\in\operatorname{graph}f_{n}$, then (3.1) follows by (3.18) and the concavity of $\eta_{n}$ on $U$.

So it is sufficient to verify (3.2). To this end, first define on $U$ the functions

[TABLE]

Since each $\operatorname{graph}\omega_{i}$ is covered by graphs of two affine functions, we see that

[TABLE]

Now consider an arbitrary $z\in U$ and choose a point $z^{*}\in\Pi(z)$. Since $d_{n}(z)\leq 1/5$ by (3.9) and $n\geq n_{0}\geq 6$, we obtain $z^{*}\in\bigcup_{1}^{n-2}[z_{i},z_{i+1}]$.

If $z^{*}=z_{i}$ for some $1\leq i\leq n-1$ with $\alpha_{i}\neq 0$, then we easily see that $z\in V_{i}$ and $d_{n}(z)=|z-z_{i}|$, and consequently

[TABLE]

where

[TABLE]

is concave on $U$.

If $z^{*}=z_{i}$ and $\alpha_{i}=0$, or $z^{*}\in[z_{i},z_{i+1}]\setminus\{z_{i},z_{i+1}\}$ for some $1\leq i\leq n-1$, then clearly $d_{n}(z)=\omega_{i}(z)$ and so $\gamma(z)=\mu_{i}(z)$.

So we have proved that the graph of $\gamma=c^{*}_{n}$ is covered by graphs of functions $\nu_{i},\ 1\leq i\leq n-1$, $\alpha_{i}\neq 0$, and functions $\mu_{i},\ 1\leq i\leq n-1$. Using (3.22), we obtain (3.2) and Lemma 3.1 implies that $\gamma=c^{*}_{n}$ is concave. ∎

4. Other results

We finish the article with a number of additional results on the systems

[TABLE]

First we observe that the description of $\cal D_{1}$ is very simple since

[TABLE]

Indeed, if the system of all components of $\varnothing\neq A\subset{\mathbb{R}}$ is locally finite, then Lemma 2.1 (ii) easily implies that $d_{A}$ is DC.

If the system of all components of $A$ is not locally finite, then there exists a sequence $(c_{n})$ of centres of components of ${\mathbb{R}}\setminus A$ converging to a point $a\in A$. Therefore $d_{A}$ is not one-sidedly strictly differentiable at $a$, since $(d_{A})^{\prime}_{\pm}(c_{n})=\mp 1$. Consequently $d_{A}$ is not DC, since each DC function on ${\mathbb{R}}$ is one-sidedly strictly differentiable at each point (see [18, Note 3.2] or [19, Proposition 3.4(i) together with Remark 3.2]).

From this characterisation easily follows that $\cal D_{1}$ is closed with respect to finite unions and intersections and that, for a closed set $M\subset\mathbb{R}$,

[TABLE]

Concerning $d\geq 2$ further observe that

[TABLE]

Indeed, if $\varnothing\neq A\in\cal D_{d}$ and $\varnothing\neq B\in\cal D_{d}$, then $d_{A\cup B}=\min(d_{A},d_{B})$ and so $d_{A\cup B}$ is DC by Lemma 2.1 (i).

Example 4.1 below shows that already $\cal D_{2}$ is not closed with respect to finite intersections. Equivalence (4.2) does not generalize already to dimension $2$ either (see again Example 4.1), however, one can see that, for a closed set $M\subset\mathbb{R}^{d}$,

[TABLE]

To prove one implication suppose $\partial M\in\cal D_{d}$. If $x\not\in M$ then clearly $\Pi_{M}(x)\in\partial M$ and so $d_{M}(x)=d_{\partial M}(x)$. Consequently, for each $x\in\mathbb{R}^{d}$, $d_{M}(x)\in\{0,d_{\partial M}(x)\}$ and so $M\in\cal D_{d}$ by Lemma 2.1 (iv). Similarly, if $x\not\in\overline{{\mathbb{R}}^{d}\setminus M}$ then $\Pi_{\overline{{\mathbb{R}}^{d}\setminus M}}(x)\in\partial(\overline{{\mathbb{R}}^{d}\setminus M})\subset\partial M$ so again $d_{\overline{{\mathbb{R}}^{d}\setminus M}}(x)\in\{0,d_{\partial M}(x)\}$ and $\overline{{\mathbb{R}}^{d}\setminus M}\in\cal D_{d}$ follows.

To prove the opposite implication it is enough to show that $d_{\partial M}=\max(d_{M},d_{\overline{{\mathbb{R}}^{d}\setminus M}})$ if $\partial M\not=\varnothing$. Clearly $d_{\partial M}\geq\max(d_{M},d_{\overline{{\mathbb{R}}^{d}\setminus M}})$, since $\partial M=M\cap\overline{{\mathbb{R}}^{d}\setminus M}$. To prove the opposite inequality suppose to the contrary that

[TABLE]

for some $x\in\mathbb{R}^{d}$. Consequently $U(x,r)\cap M\not=\varnothing$ and $U(x,r)\cap\overline{{\mathbb{R}}^{d}\setminus M}\not=\varnothing$. Then also $U(x,r)\cap{\mathbb{R}}^{d}\setminus M\not=\varnothing$ and thus $U(x,r)\cap\partial M\not=\varnothing$ which is a contradiction.

Before presenting the following example we first observe that the function $g(x)=x^{5}\cos\frac{\pi}{x}$, $x\not=0$, $g(0)=0$, is $\mathcal{C}^{2}$ on $\mathbb{R}$ and therefore DC by Lemma 2.1 (v). Indeed, a direct computation shows that $g^{\prime\prime}(x)=x\left(8\pi x\sin\frac{\pi}{x}-(\pi^{2}-20x^{2})\cos\frac{\pi}{x}\right)$, $x\not=0$, and $g^{\prime\prime}(0)=0$.

Example 4.1**.**

There are sets $A,B\in\cal D_{2}$ such that $A\cap B\not\in\cal D_{2}$. Further, there is a set $K\in{\cal D}_{2}$ such that $\partial K\not\in{\cal D}_{2}$ and $\overline{\mathbb{R}^{2}\setminus K}\not\in{\cal D}_{2}$.

Proof.

Define $g(x)=x^{5}\cos\frac{\pi}{x}$, $x\not=0$, $g(0)=0$, and $f(x)=0$, $x\in\mathbb{R}$. Put

[TABLE]

Since both $f$ and $g$ are DC, we obtain that $A,B\in\cal D_{2}$ by Theorem 3.3 and (4.4). Put $M=A\cap B$ and $K=\overline{\mathbb{R}^{2}\setminus M}$. Clearly also $M=\overline{\mathbb{R}^{2}\setminus K}$. First note that $M\not\in\cal D_{2}$ since the function $x\mapsto d_{M}(x,0)$ is equal to $d_{H}$, but clearly $H\not\in\cal D_{1}$ by(4.1).

We obtain that $K\in\cal D_{2}$ by (4.3), since $K=C\cup D$, where $C=\{(x,y):y\leq f(x)\}$ and $D=\{(x,y):y\geq g(x)\}$, and $C,D\in\cal D_{2}$ by Theorem 3.3 and (4.4). Finally, $\partial K\not\in\cal D_{2}$ by (4.4) applied to $K$. ∎

Now we will show that equivalence (4.2) holds for sets $M$ of positive reach (cf. (4.5)). We first recall their definition.

If $A\subset{\mathbb{R}}^{d}$ and $a\in A$, we define

[TABLE]

and the reach of $A$ as

[TABLE]

Note that each set with positive reach is clearly closed.

As mentioned in Introduction, it is essentially well-known that

[TABLE]

Indeed, for each $a\in A$ [5, Proposition 5.2] implies that $d_{A}$ is semiconvex on $U(a,{\rm reach}\,A/2)$, which with (2.2) and Lemma 2.1 (ii) implies that $d_{A}$ is DC.

Proposition 4.2**.**

Let $\varnothing\neq A\subset{\mathbb{R}}^{d}$ be a set with positive reach and $B\coloneqq\overline{{\mathbb{R}}^{d}\setminus A}$. Then both $B$ and $\partial A$ belong to $\cal D_{d}$.

Proof.

By (4.5) and (4.4) it is sufficient to prove that $B\in\cal D_{d}$. Since $d_{B}$ is locally DC on ${\mathbb{R}}^{d}\setminus B$ (see (2.2)) and on $\mathrm{int}B$ (trivially), by Lemma 2.1 (ii) it is sufficient to prove that

[TABLE]

To prove (4.6), choose $0<r<{\rm reach}\,A$ and denote $A_{r}\coloneqq\{x:\ {\rm dist}\,(x,A)=r\}$. We will first prove that

[TABLE]

To this end, choose an arbitrary $x\in\mathrm{int}A$. Obviously, there exists $y\in\partial B\subset\partial A$ such that ${\rm dist}\,(x,B)=|x-y|$. Since $A$ has positive reach and $y\in\partial A$, there exists $z\in A_{r}$ such that $|y-z|=r$ (It follows, e.g., from [14, Proposition 3.1 (v),(vi)]). Therefore

[TABLE]

To prove the opposite inequality, choose a point $z^{*}\in A_{r}$ such that ${\rm dist}\,(x,A_{r})=|x-z^{*}|$. Obviously, on the segment $[x,z^{*}]$ there exists a point $y^{*}\in\partial A\subset B$. Then

[TABLE]

and (4.7) is proved.

Now let $a\in\partial B\subset\partial A$ be given. Then $a\notin A_{r}$ and so by (2.2) there exists $\rho>0$ such that ${\rm dist}\,(\cdot,A_{r})$ is DC on $U(a,\rho)$. For $x\in U(a,\rho)$, $d_{B}(x)={\rm dist}\,(x,A_{r})-r$ if $x\in\mathrm{int}A$ (by (4.7)) and $d_{B}(x)=0$ if $x\notin\mathrm{int}A$. Thus Lemma 2.1 (iv) implies that $d_{B}$ is DC on $U(a,\rho)$, which proves (4.6). ∎

Further recall that our main result (Theorem 3.3) asserts that

[TABLE]

Motivated by a natural question, for which non DC functions $g$ (4.8) holds, we present the following result, whose proof is implicitly contained in the proof of [12, Proposition 6.6]; see Remark 4.4 below.

Proposition 4.3**.**

If $g:{\mathbb{R}}^{d-1}\to{\mathbb{R}}$ ($d\geq 2$) is locally Lipschitz and $A\coloneqq\operatorname{graph}g\in\cal D_{d}$, then $g$ is DC.

Remark 4.4*.*

One implication of [12, Proposition 6.6] gives that if $A$ is as in Proposition 4.3 (or, more generally, $A$ is a Lipschitz manifold of dimension $0<k<d$; see [12, Definition 2.4] for this notion) and $A$ is WDC, then $g$ is DC (or is a DC manifold of dimension $0<k<d$, respectively). The proof of this implication works with an aura $f=f_{M}$ of a set $M$, but under the assumption that $A\in\cal D_{d}$, the proof clearly also works, if we use the distance function $d_{A}$ instead of $f$. So we obtain not only Proposition 4.3, but also the following more general result.

If $A\subset{\mathbb{R}}^{d}$ is a Lipschitz manifold of dimension $0<k<d$ and $A\in\cal D_{d}$, then $A$ is a DC manifold of dimension $k$.

Recall that it is an open question, whether $\operatorname{graph}g\in\cal D_{d}$, whenever $g:\mathbb{R}^{d-1}\to\mathbb{R}$ is a DC function. However, using Proposition 4.2, we easily obtain:

Corollary 4.5**.**

If $g:\mathbb{R}^{d-1}\to\mathbb{R}$ is a semiconcave function then $\operatorname{graph}g\in\cal D_{d}$.

Proof.

The set $S\coloneqq\{(a,b)\in\mathbb{R}^{d-1}\times\mathbb{R}:b\leq g(a)\}$ has positive reach by [6, Theorem 2.3] and consequently $d_{\operatorname{graph}g}=d_{\partial S}$ is DC by Proposition 4.2. ∎

Remark 4.6**.**

Let $M\subset\mathbb{R}^{d}$ be a closed set whose boundary can be locally expressed as a graph of a semiconvex function (i.e., for each $a\in\partial M$ there exist a semiconcave function $g:\mathbb{R}^{d-1}\to\mathbb{R}$, $\delta>0$ and an isometry $\varphi:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}$ such that $\partial M\cap U(a,\delta)=\varphi(\operatorname{graph}g)\cap U(a,\delta)$). Then $d_{\partial M}$ is locally DC (and therefore DC) by Corollary 4.5 and (2.2) and so $M\in\cal D_{d}$ by Lemma 2.1 (iv) and (4.4).

Before the next results, we present the following definitions: we say that a set $A\subset\mathbb{R}^{d}$ is a DC hypersurface, if there exist a vector $v\in S^{d-1}$ and a DC function (i.e. the difference of two convex functions) $g$ on $W\coloneqq(\operatorname{span}v)^{\bot}$ such that $A=\{w+g(w)v:w\in W\}$. A set $P\subset\mathbb{R}^{2}$ will be called a DC graph if it is a rotated copy of $\operatorname{graph}(f|_{I})$ for a DC function $f:\mathbb{R}\to\mathbb{R}$ and some compact (possibly degenerated) interval $\varnothing\not=I\subset\mathbb{R}$. Note that $P$ is a DC graph if and only if it is a nonempty connected compact subset of a DC hypersurface in $\mathbb{R}^{2}$.

Proposition 4.7**.**

Let $d\geq 2$ and $F\in\cal D_{d}$. Then each bounded set $C\subset\partial F$ can be covered by finitely many DC hypersurfaces.

Proof.

By our assumptions, $f\coloneqq{\rm dist}\,(\cdot,F)$ is a DC function on ${\mathbb{R}}^{d}$ and $f(x)=0$ for every $x\in C$. So, by [12, Crollary 5.4] it is sufficient to prove that for each $x\in C$ there exists $y^{*}\in\partial f(x)$ with $|y^{*}|>\varepsilon\coloneqq 1/4$, where $\partial f(x)$ is the Clarke generalized gradient of $f$ at $x$ (see [3, p. 27]). To this end, suppose to the contrary that $x\in C$ and $\partial f(x)\subset B(0,1/4)$. Since the mapping $x\mapsto\partial f(x)$ is upper semicontinuous (see [3, Proposition 2.1.5 (d)]), there exists $\delta>0$ such that $\partial f(u)\subset U(0,1/2)$ for each $u\in U(x,\delta)$. Since $x\in\partial F$, we can choose $z\in U(x,\delta/2)\setminus F$ and $p\in\Pi_{F}(z)$. Then $p\in U(x,\delta)$, $f(z)-f(p)=|z-p|$ and Lebourg’s mean-value theorem (see [3, Theorem 2.3.7]) implies that there exist $u\in U(x,\delta)$ and $\alpha\in\partial f(u)$ such that

[TABLE]

Therefore $|\alpha|\geq 1$, which is a contradiction. ∎

The above proposition easily implies the following fact.

Corollary 4.8**.**

If $F\in\cal D_{2}$ then $\partial F$ is a subset of the union of a locally finite system of DC graphs.

Using Theorem 3.3 we obtain the following easy result.

Proposition 4.9**.**

*If $A\subset\mathbb{R}^{2}$ is the union of a locally finite system of DC graphs then $A\in\cal D_{2}$. *

Proof.

First note that it is enough to prove that any DC graph $P$ belongs to $\cal D_{2}$. Indeed, if $M$ is a locally finite system of DC graphs and each DC graph belongs to $\cal D_{2}$, then $d_{M}$ is locally DC by (4.3) (and so DC) and $M\in\cal D_{2}$.

So assume that $A$ is a DC graph. Without any loss of generality we may assume that $A=\operatorname{graph}f|_{[0,p]}$ for some DC function $f:\mathbb{R}\to\mathbb{R}$. If $p=0$ then $d_{A}=|\cdot|$ is even convex, so assume that $p>0$. We may also assume that $f(0)=0$.

First note that (by Theorem 3.3 and (2.2)) $d_{A}$ is locally DC on $\mathbb{R}^{2}\setminus\{(0,0),(p,f(p))\}$. It remains to prove that $d_{A}$ is DC on some neighbourhood of $(0,0)$ and $(p,f(p))$. We will prove only the case of the point $(0,0)$, the other case can be proved quite analogously. By Lemma 2.1 (iii) we can choose $L>0$ such that $f$ is $L$-Lipschitz on $[0,p]$. Define

[TABLE]

It is easy to see that both $f_{+}$ and $f_{-}$ are continuous and so 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 Theorem 3.3, $d_{0}$ and $d_{3}$ are even convex 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. To finish the proof it is enough to observe that $d_{A}=\tilde{d}$ on $U(0,\frac{p}{2})$. ∎

However, the following example shows that the opposite implication does not hold even for nowhere dense sets $A$.

Example 4.10**.**

There is a nowhere dense set $A\in{\cal D}_{2}$ which is not the union of a locally finite system of DC graphs.

Proof.

Define

[TABLE]

Put $A^{\pm}\coloneqq\operatorname{graph}(\pm f)$, $A_{k}\coloneqq\operatorname{graph}g_{k}$, $k\in\mathbb{N}$, and

[TABLE]

$A$ is clearly closed and nowhere dense, and it is not the union of a locally finite system of DC graphs since every DC graph $B\subset A$ can intersect at most one of the sets $A_{i}$. It remains to prove that $A\in{\cal D}_{2}$.

First we will describe all components of ${\mathbb{R}}^{2}\setminus A$. To this end, for each $k\in\mathbb{N}$, define

[TABLE]

[TABLE]

Set $G_{k}\coloneqq\{(x,y):\ L_{k}(x)<y<U_{k}(x)\}$, $k=0,1,2,\dots$. Then

[TABLE]

are clearly all components of ${\mathbb{R}}^{2}\setminus A$.

Recall that both $U_{k}$ and $L_{k}$ is defined on $D_{k}$, where $D_{k}=[\frac{1}{2k+3},\frac{1}{2k}]$ for $k\in\mathbb{N}$ and $D_{0}=[1/3,\infty)$. Using the facts that $D_{k}$ and $D_{k+2}$ are disjoint ($k=0,1,\dots$),

[TABLE]

and $U_{0}(1/3)=L(1/3)=g(1/3)$, it is easy to see that there exist unique functions $U$, $\tilde{U}$ which are continuous on ${\mathbb{R}}$, $U$ (resp. $\tilde{U}$) extends all $U_{k}$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) and $U$ (resp. $\tilde{U}$) equals to $g$ at all points at which no $U_{k}$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) is defined. Quite analogously a continuous function $L$ (resp. $\tilde{L}$) extending all $L_{k}$, $k=0,2,4,\dots$ (resp. $k=1,3,5,\dots$) is defined. Since the functions $g$, $f$, $-f$ are DC, Lemma 2.1 (iv) implies that the functions $U$, $\tilde{U}$, $L$, $\tilde{L}$ are DC. So Theorem 3.3 implies that the distance functions

[TABLE]

are DC.

Obviously $d_{A}(x)=0$ for $x\in A$, $d_{A}(x)=d_{A^{+}}(x)$ for $x\in G^{+}$ and $d_{A}(x)=d_{A^{-}}(x)$ for $x\in G^{-}$. Further, if $x\in G_{k}$ with $k=2,4,6,\dots$, then

[TABLE]

which easily follows from the facts that

[TABLE]

Similarly we obtain that, if $x\in G_{k}$ with $k=1,3,5,\dots$, then

[TABLE]

Thus, using (4.10) and Lemma 2.1 (iv), we obtain that $d_{A}$ is DC. ∎

It seems that there does not exist an essentially simpler example. Iterating the construction of the example we can obtain nowhere dense sets in $\cal D_{2}$ of quite complicated topological structure.

In our opinion, using Proposition 4.7 and Theorem 3.3 it is possible to give an optimal complete characterisation of sets in $\cal D_{2}$, but it appears to be a rather hard task. We believe that we succeeded to find some characterisation, however, it is not quite satisfactory and our current proof is very technical. We aim to find a better characterisation, hopefully with a simpler proof.