D-CLOSE Sets and DC Distance Functions

• D-CLOSE sets are closed subsets in ℝᵈ whose distance function can be decomposed as the difference of two convex functions, capturing subtle geometric regularity.
• They encompass planar DC graphs and higher-dimensional semiconcave graphs, illustrating how analytic structure and geometric complexity coexist.
• Key findings include a complete 1D characterization, stability results, and counterexamples showing that D-CLOSE properties are not generally preserved under intersections.

Searching arXiv for papers on D-CLOSE sets, DC distance functions, and related positive-reach/WDC literature. D-CLOSE denotes the class of nonempty closed sets $F\subset \mathbb{R}^d$ whose distance function

$d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$

is a DC function on $\mathbb{R}^d$, that is, a difference of two convex functions. The systematic study of such sets in Euclidean space was developed in "On sets in ${\mathbb R}^d$ with DC distance function" (Pokorný et al., 2019). The central theme is that, although $(d_F)^2$ is always DC, the unsquared distance $d_F$ need not be DC, so the D-CLOSE property isolates a nontrivial geometric class. The paper establishes a complete criterion in dimension one, proves that every planar graph of a DC function is D-CLOSE, extends this in higher dimensions to graphs of semiconcave locally Lipschitz functions, and exhibits both stability phenomena and sharp counterexamples (Pokorný et al., 2019).

1. Definition and basic analytic framework

A function $f:\mathbb{R}^d\to\mathbb{R}$ is DC if it admits a decomposition

$f(x)=f_1(x)-f_2(x),$

where $f_1,f_2$ are convex on $\mathbb{R}^d$. On an open convex set $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$0, the related notions of semiconvexity and semiconcavity are formulated by requiring that $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$1 or $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$2, respectively, for some convex $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$3 and $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$4 (Pokorný et al., 2019).

The class of DC functions has several closure properties used throughout the theory. Linear combinations, pointwise maxima and minima, and absolute values of DC functions remain DC. A function that is locally DC on a convex domain is globally DC. DC functions are locally Lipschitz. If a continuous function equals one of finitely many DC functions at each point, then it is DC. Every $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$5 function is DC (Pokorný et al., 2019).

For a nonempty closed set $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$6, the distance function $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$7 is always well defined and globally Lipschitz. However, the distinction between $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$8 and $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$9 is essential: the latter is always DC, while the former may fail to be DC (Pokorný et al., 2019). This motivates the notation

$\mathbb{R}^d$0

A D-CLOSE set is therefore a geometric object whose distance-to-set function retains the structural regularity of DC analysis.

This framework places D-CLOSE sets at the intersection of nonsmooth analysis, convexity-based decomposition theory, and geometric measure theory. A plausible implication is that D-CLOSE regularity is strong enough to encode geometric information beyond mere metric regularity, but weak enough to include sets substantially more singular than smooth submanifolds.

2. Complete characterization in one dimension

In dimension one, the class $\mathbb{R}^d$1 admits an exact geometric characterization. A closed set $\mathbb{R}^d$2 belongs to $\mathbb{R}^d$3 if and only if the family of connected components of $\mathbb{R}^d$4 is locally finite in $\mathbb{R}^d$5 (Pokorný et al., 2019).

This criterion is both necessary and sufficient. If $\mathbb{R}^d$6 has only finitely many gaps near each point, then $\mathbb{R}^d$7 is locally semiconvex on $\mathbb{R}^d$8, vanishes identically on $\mathbb{R}^d$9, and can be assembled globally using the closure properties of DC functions under local-to-global passage and finite mixing (Pokorný et al., 2019). Conversely, if gaps accumulate at a point ${\mathbb R}^d$0, then ${\mathbb R}^d$1 fails one-sided strict differentiability at ${\mathbb R}^d$2, whereas DC functions on ${\mathbb R}^d$3 are one-sided strictly differentiable everywhere; hence ${\mathbb R}^d$4 cannot be DC (Pokorný et al., 2019).

The one-dimensional theory yields several immediate structural consequences. The class ${\mathbb R}^d$5 is stable under finite unions and intersections. The exposition also records the equivalence

${\mathbb R}^d$6

for closed ${\mathbb R}^d$7 (Pokorný et al., 2019). Taken together, these properties show that ${\mathbb R}^d$8 is governed by a purely local combinatorial restriction on connected components rather than by curvature or smoothness assumptions.

The one-dimensional criterion serves as a baseline for higher-dimensional questions. It suggests that the obstruction to DC regularity of the distance function is tied to geometric accumulation of “gaps,” but the higher-dimensional analogues are substantially more intricate.

3. Planar DC graphs

The main theorem of the paper concerns graphs in ${\mathbb R}^d$9. If $(d_F)^2$0 is a DC function, then its graph

$(d_F)^2$1

belongs to $(d_F)^2$2; equivalently, the map

$(d_F)^2$3

is DC on $(d_F)^2$4 (Pokorný et al., 2019).

The proof proceeds by local analysis near points of the graph. One writes $(d_F)^2$5 with $(d_F)^2$6 convex and Lipschitz on a bounded interval, then approximates $(d_F)^2$7 uniformly by piecewise-linear functions $(d_F)^2$8. The corresponding distance functions $(d_F)^2$9 converge to the distance $d_F$0 from the graph of $d_F$1. On a neighborhood of any graph point, one constructs a concave corrector $d_F$2, uniformly Lipschitz in $d_F$3, such that $d_F$4 is concave on that neighborhood. This step uses a mixing lemma for concave patches together with an angular-sector construction in which local expressions of the form $d_F$5 become affine on each sector. Arzelà–Ascoli then yields a subsequential limit of the correctors, giving local DC regularity of $d_F$6, and a patching argument promotes this to global DC regularity (Pokorný et al., 2019).

The theorem identifies planar DC graphs as a broad and nontrivial source of D-CLOSE sets. Since every $d_F$7 function is DC, graphs of $d_F$8 planar curves fall under the result, but the theorem is stronger: it applies to arbitrary DC graphs, which may have corners or other nonsmooth features compatible with DC structure (Pokorný et al., 2019).

This result is specific to dimension two in its present form. The paper emphasizes that the corresponding statement for arbitrary DC graphs in higher dimensions remains unresolved (Pokorný et al., 2019). This dimensional asymmetry is one of the main unresolved aspects of D-CLOSE theory.

4. Higher-dimensional semiconcave graphs and related positive results

For dimensions $d_F$9, the paper proves a higher-dimensional analogue under a stronger hypothesis. If

$f:\mathbb{R}^d\to\mathbb{R}$0

is semiconcave and locally Lipschitz, then its graph

$f:\mathbb{R}^d\to\mathbb{R}$1

belongs to $f:\mathbb{R}^d\to\mathbb{R}$2 (Pokorný et al., 2019).

The argument passes through sets of positive reach. A standard result asserts that the epigraph $f:\mathbb{R}^d\to\mathbb{R}$3 of a semiconcave $f:\mathbb{R}^d\to\mathbb{R}$4 has positive reach. Any set of positive reach lies in $f:\mathbb{R}^d\to\mathbb{R}$5, so the distance to the epigraph is DC. A boundary-mixing argument then yields the DC property for the distance to the boundary, which is precisely the graph $f:\mathbb{R}^d\to\mathbb{R}$6 (Pokorný et al., 2019).

The paper records several further positive classes of D-CLOSE sets. Every convex body belongs to $f:\mathbb{R}^d\to\mathbb{R}$7, and so does every finite union of convex bodies. More generally, every set of positive reach is D-CLOSE. In addition, if $f:\mathbb{R}^d\to\mathbb{R}$8 is a DC hypersurface, locally representable as a rotated graph of a DC function $f:\mathbb{R}^d\to\mathbb{R}$9, then $f(x)=f_1(x)-f_2(x),$0; the paper also notes a WDC-manifold strengthening (Pokorný et al., 2019).

The following table summarizes the principal positive results stated in the source.

Class of sets	Ambient dimension	D-CLOSE status
Closed $f(x)=f_1(x)-f_2(x),$1 with locally finite connected components	$f(x)=f_1(x)-f_2(x),$2	Necessary and sufficient criterion
Graphs $f(x)=f_1(x)-f_2(x),$3 of DC functions $f(x)=f_1(x)-f_2(x),$4	$f(x)=f_1(x)-f_2(x),$5	In $f(x)=f_1(x)-f_2(x),$6
Graphs $f(x)=f_1(x)-f_2(x),$7 of semiconcave locally Lipschitz $f(x)=f_1(x)-f_2(x),$8	$f(x)=f_1(x)-f_2(x),$9	In $f_1,f_2$0
Sets of positive reach	any $f_1,f_2$1	In $f_1,f_2$2
Convex bodies and finite unions of convex bodies	any $f_1,f_2$3	In $f_1,f_2$4
DC hypersurfaces	any $f_1,f_2$5	In $f_1,f_2$6

These results situate D-CLOSE sets within a wider family of geometrically regular objects while showing that the class is not limited to classical smooth or convex categories.

5. Examples, counterexamples, and non-closure phenomena

The paper gives examples demonstrating that $f_1,f_2$7 has more subtle behavior than standard regularity classes. One counterexample shows that $f_1,f_2$8 is not closed under intersection. Let

$f_1,f_2$9

where both functions are DC on $\mathbb{R}^d$0. Define

$\mathbb{R}^d$1

Then $\mathbb{R}^d$2 by the planar graph theorem, but

$\mathbb{R}^d$3

fails to have DC distance because its vertical gap structure oscillates infinitely and accumulates at the $\mathbb{R}^d$4-axis; the distance function even fails one-sided differentiability at infinitely many points (Pokorný et al., 2019).

This example is important because it separates D-CLOSE regularity from naive algebraic stability. In one dimension, finite intersections behave well; in dimension two, they do not. The paper therefore shows that higher-dimensional D-CLOSE geometry cannot be understood simply by extrapolating the one-dimensional criterion.

A different example shows that D-CLOSE sets can be highly non-manifold-like. A Cantor-like union of tiny DC graphs $\mathbb{R}^d$5 can be constructed so that

$\mathbb{R}^d$6

is nowhere dense and belongs to $\mathbb{R}^d$7, yet cannot be covered by any locally finite family of DC graphs (Pokorný et al., 2019). The mechanism is that each complementary component lies between two adjacent DC graphs, so on each open strip the distance is the minimum of two DC functions; globally, finitely many such patches can be mixed (Pokorný et al., 2019).

These examples show that D-CLOSE sets may be sparse, oscillatory, and globally complicated while still admitting a DC distance function. At the same time, the class is delicate under set-theoretic operations.

6. Open problems and mathematical significance

The paper identifies the main open problem as the higher-dimensional DC-graph question: if $\mathbb{R}^d$8 is an arbitrary DC function, is its graph $\mathbb{R}^d$9 in $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$00 for $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$01? The answer is affirmative for $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$02, and also affirmative in all dimensions when $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$03 is semiconcave, but it remains unclear for general DC $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$04 in dimensions $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$05 (Pokorný et al., 2019).

A second direction concerns a complete geometric characterization of planar D-CLOSE sets. The authors indicate that there should be a full structural description of sets in $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$06 whose distance function is DC, potentially involving DC graphs and positive-reach arcs, but a concise final formulation is not yet available. The nowhere-dense example shows that neither nowhere denseness nor local finiteness of families of DC graphs is sufficient as a simple criterion (Pokorný et al., 2019).

A third theme is the algebra of $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$07. Beyond finite unions, the class is not closed under intersections in dimensions $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$08, and the paper raises the question of stability under other natural operations such as Minkowski addition or Cartesian products (Pokorný et al., 2019).

The significance of these questions lies in the position of D-CLOSE theory between analytic decomposability and geometric structure. A plausible implication is that a full characterization of $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$09 or of higher-dimensional DC graphs would provide a new bridge between DC analysis, singularity theory, and the metric geometry of nonsmooth sets.

7. Relation to neighboring regularity classes

D-CLOSE sets should be distinguished from several adjacent notions. They are not defined by smoothness, even though every $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$10 function is DC and smooth graphs supply examples. They are not defined by convexity, even though convex bodies and finite unions of convex bodies are included. They are not equivalent to positive-reach sets, since the planar DC-graph theorem covers sets that are not presented as positive-reach objects, and the nowhere-dense example shows that D-CLOSE geometry can be substantially more irregular (Pokorný et al., 2019).

The relation with DC hypersurfaces is particularly close. The paper states that if a set is locally a rotated graph of a DC function, then it lies in $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$11, and mentions a strengthening for WDC manifolds (Pokorný et al., 2019). This places D-CLOSE theory in the broader context of weakly DC geometry. However, the failure of intersection closure and the existence of complicated nowhere-dense examples show that the class of all D-CLOSE sets is larger and less rigid than a simple manifold-based taxonomy would suggest.

From an analytic viewpoint, the D-CLOSE property is best understood as a statement about the structural complexity of the metric projection landscape encoded by $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$12. When $d_F(x)=\mathrm{dist}(x,F)=\inf_{y\in F}\|x-y\|$13 is DC, one gains access to the machinery of DC calculus and to local Lipschitz control, but the examples in (Pokorný et al., 2019) show that this regularity coexists with substantial geometric complexity. That tension between analytic tractability and geometric richness is the defining feature of the subject.