Union-Ball Formulation in Geometry

• Union-ball formulation is a geometric representation method for closed sets meeting an interior sphere condition, leveraging equal-radius ball unions.
• It analyzes both weak and strong forms of the conjecture, with the planar case optimally proven at radius r/√3.
• The formulation informs diverse areas like nonsmooth analysis, control theory, and computational geometry by offering a precise structure theorem.

Searching arXiv for the primary paper and a small set of related papers to ground the article. {"^^^^^^^^2query2 OR ti:\2^^^^^^^^"Validity of the Strong Version of the Union of Uniform Closed Balls Conjecture in the Plane\"","max_results":5} {"^^^^^^^^2query2 formulation\" arXiv","max_results":^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^} The union-ball formulation is a geometric representation principle for closed sets with a uniform interior sphere condition. In its primary usage, one works in Euclidean space PRESERVED_PLACEHOLDER_^^^^2query2^^^^ and asks whether a nonempty closed set PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^ satisfying an interior $r$-sphere condition can be recovered exactly as the union of all closed balls of a single radius $r'$ contained in $S$. Writing

$$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$

the formulation is the equality $S=S_{r'}$. The recent planar result proves the strong version of this statement with the optimal radius $r/\sqrt{3}$, thereby resolving the two-dimensional case of the strong union of uniform closed balls conjecture (&&&^^^^2query2^^^^&&&).

^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^. Formal definition and geometric setting

The ambient space is $\mathbb{R}^n$ with the usual inner product $\langle\cdot,\cdot\rangle$ and norm PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^. For PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^ and PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\22^^^^, PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^3 denotes the closed ball, PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^4 the open ball, and PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^5 the sphere. For a set PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^6, the notation PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^7, PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^8, PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^9, and $r$^^^^2query2^^^^ refers respectively to the interior, closure, boundary, and complement; the paper also uses

$r$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^

A nonempty closed set $r$2 satisfies the interior $r$3-sphere condition if for every $r$4 there exists $r$5 such that

$r$6

This is a local rolling-ball condition from the inside. The paper records an equivalent proximal-normal formulation: $r$7 is regular closed, and for each $r$8 there exists a unit proximal normal $r$9 realized by an $r'$^^^^2query2^^^^-sphere, equivalently

$r'$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^

A set $r'$2 is a union of uniform closed balls of radius $r'$3 if

$r'$4

for some index set $r'$5. The radius is common to all balls; there is no restriction on cardinality, overlap, or center placement beyond exact reconstruction of $r'$6. In this language, the union-ball formulation asks for a radius $r'$7 depending only on $r'$8 and the dimension such that

$r'$9

This is an exact structure theorem, not a volumetric or extremal inequality (&&&^^^^2query2^^^^&&&).

2. Weak and strong forms of the conjecture

The literature distinguishes a weak and a strong version. The weak version asks only for existence of some common inscribed radius $S$^^^^2query2^^^^. As summarized in the planar paper, a general theorem already shows that one may always take $S$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^ in every dimension, and that no larger radius can be uniformly valid in all dimensions for all such sets. The strong version specifies a dimension-dependent candidate

$S$2

motivated by examples showing that, in dimension $S$3, radii $S$4 fail in general (&&&^^^^2query2^^^^&&&).

Statement	Radius in the conclusion	Status
Weak version	some $S$5	subsumed by the $S$6 theorem
General sharp theorem	$S$7	valid in all dimensions
Strong version	$S$8	proved in the plane
Planar value	$S$9	optimal in $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$^^^^2query2^^^^

For $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^, the strong radius becomes

$$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$2

The planar theorem therefore verifies the strong conjecture with the expected constant. Conceptually, the weak statement is existential, whereas the strong statement is quantitative and dimension-sensitive.

3. Structural content of the formulation

Geometrically, the hypothesis is local: every boundary point admits an interior tangent ball of fixed radius $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$3. The conclusion is global: every point of $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$4 lies in some interior ball of a single radius $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$5. The quantities singled out in the paper are the boundary distance

$$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$6

for $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$7, and the configuration of proximal normals $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$8 along the boundary.

This is why the union-ball formulation is described as a structure theorem. It does not compare areas, volumes, or perimeters. Instead, it asks whether the local interior sphere condition forces an exact global representation by equal-radius balls. In that sense, the strong conjecture asserts a sharp global inner-thickness estimate: if every boundary point admits an interior ball of radius $$S_{r'}:=\bigcup\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\},$$9, then every point of $S=S_{r'}$^^^^2query2^^^^ lies in an interior ball of radius

$S=S_{r'}$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^

and larger uniform radii cannot generally work in dimension $S=S_{r'}$2.

The paper also places the formulation in the language of nonsmooth analysis. The interior sphere condition is equivalent to proximal smoothness, and the representation $S=S_{r'}$3 supplies a constructive geometric description of such sets. There is an indirect relation to Minkowski sums, tubular neighborhoods, reach, and rolling-ball conditions, because unions of equal-radius balls arise naturally in those settings; however, the conjecture itself is not stated as a measure inequality or a rearrangement principle (&&&^^^^2query2^^^^&&&).

4. The planar theorem and the proof architecture

The main theorem states that if $S=S_{r'}$4 is nonempty, closed, and satisfies the interior $S=S_{r'}$5-sphere condition, then

$S=S_{r'}$6

Equivalently, $S=S_{r'}$7. This proves the strong version in the plane with the optimal constant (&&&^^^^2query2^^^^&&&).

The proof is formulated entirely in union-ball terms. For a candidate radius $S=S_{r'}$8, define

$S=S_{r'}$9

One argues by contradiction, assuming $r/\sqrt{3}$^^^^2query2^^^^. A point $r/\sqrt{3}$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^ is chosen, together with a nearest boundary point $r/\sqrt{3}$2, and

$r/\sqrt{3}$3

Since $r/\sqrt{3}$4 is not covered by an interior ball of the target radius, one obtains $r/\sqrt{3}$5.

A core local result, Lemma 3.2, shows that $r/\sqrt{3}$6 is non-regular: the proximal normal cone is not a single ray. More precisely, there exist two distinct unit proximal normals $r/\sqrt{3}$7, both realized by radius-$r/\sqrt{3}$8 balls, and they satisfy a system of angular inequalities. Their effect is to force a wide cone of inward normals at $r/\sqrt{3}$9, together with a tightly constrained location of the direction $\mathbb{R}^n$^^^^2query2^^^^ toward $\mathbb{R}^n$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^. The proof then performs an extremal-ball construction along the ray $\mathbb{R}^n$2, producing further non-regular boundary points $\mathbb{R}^n$3 on the boundary of a common interior ball.

The distinctly planar ingredient is Lemma 3.^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^, a purely two-dimensional geometric lemma about one small circle and two equal larger circles. If

$\mathbb{R}^n$4

then certain pairs of points on the small circle that avoid both large circles subtend an angle strictly less than $\mathbb{R}^n$5. After translating this configuration back to the boundary geometry of $\mathbb{R}^n$6, the argument yields

$\mathbb{R}^n$7

Summing gives a total strictly less than $\mathbb{R}^n$8, contradicting the planar triangle angle sum. Hence no such $\mathbb{R}^n$9 can exist, and $\langle\cdot,\cdot\rangle$^^^^2query2^^^^.

The proof is therefore not merely existential: it identifies the obstruction to coverage by radius-$\langle\cdot,\cdot\rangle$^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^ balls and rules it out using planar angle ordering and the exact triangle identity. The authors explicitly note that this final step has no direct higher-dimensional analogue (&&&^^^^2query2^^^^&&&).

5. Consequences, analytic interpretation, and higher-dimensional status

The planar result settles the strong version of the union of uniform closed balls conjecture in dimension two. It yields a sharp description of planar sets with uniform interior disk condition: local realizability by radius-$\langle\cdot,\cdot\rangle$2 tangent disks forces global representability as a union of radius-$\langle\cdot,\cdot\rangle$3 disks. Since the constant agrees with the dimension-dependent candidate $\langle\cdot,\cdot\rangle$4 at $\langle\cdot,\cdot\rangle$5, the theorem is optimal.

From the viewpoint of proximal and variational analysis, the result refines the internal geometry of proximally smooth sets in $\langle\cdot,\cdot\rangle$6. The paper identifies applications or relevance for control theory and reflected dynamics, optimization over nonconvex but proximally smooth sets, and regularity and approximation properties of distance functions.

The higher-dimensional situation remains unresolved. The authors indicate that the early part of the argument should generalize partially: in $\langle\cdot,\cdot\rangle$7, one expects an interior ball whose boundary contains at least $\langle\cdot,\cdot\rangle$8 non-regular contact points. What is missing is a replacement for the planar angle-based contradiction. In higher dimensions, normal cones are more intricate, and there is no direct analogue of the scalar triangle-angle-sum mechanism that closes the planar proof. Accordingly, the strong version remains open for $\langle\cdot,\cdot\rangle$9, and the paper states that no proof or counterexample is known in those dimensions (&&&^^^^2query2^^^^&&&).

6. Other uses of “union-ball” language in the literature

The expression is not unique to the interior-sphere conjecture. In other areas it denotes distinct, though geometrically related, constructions.

In numerical integration, a union-of-balls domain may mean a finite multibubble

PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2query2^^^^

with QMC sampling on the solid and on the true boundary

PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^

That formulation supports Tchakaloff-like compression of QMC volume and surface integration on unions of balls (Elefante et al., 2023).

In topological data analysis, the localized union-of-balls bifiltration fixes a center PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query22^^^^ and studies

PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^3

together with a relative variant PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^4. Here the phrase refers to a two-parameter refinement of the classical union-of-balls filtration used in persistent homology (Kerber et al., 2023).

In computational geometry, the coverage question “is a finite intersection of balls covered by a finite union of balls?” is reformulated by introducing Voronoi-like polyhedra and solving associated quadratic programs. In that usage, the union-ball perspective is an exact algorithmic representation of a nonconvex coverage problem (&&&^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^&&&).

In adversarial robustness for piecewise-linear neural networks, the relevant geometry is not a union of Euclidean balls but the radius of the largest PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^5-ball centered at PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^6 that fits inside a decision region represented as a union of polytopes. The paper explicitly frames pointwise robustness as fitting a centered ball into a non-convex polyhedral region and computes the exact radius by searching the boundary of that union (&&&^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^&&&).

A complementary notion appears in the study of ball-polyhedra, where one takes finite intersections of congruent balls rather than unions: PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^7 That literature emphasizes the duality between unions and intersections under complements and studies rigidity of the resulting intersection-of-balls structures (&&&^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\22^^^^&&&).

Finally, in abstract ball spaces, “closure under unions of chains” defines a different order-theoretic completion: PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^8 There the issue is not Euclidean representation by equal-radius balls but stability of spherical completeness under chain-union closure (&&&^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2^^^^3&&&).

These usages are related by a common reliance on balls as primitive geometric objects, but they refer to different mathematical problems. In the strict sense established by the interior-sphere literature, the union-ball formulation is the exact representation problem PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^9 for sets satisfying a uniform interior sphere condition, with the planar strong theorem now settled at radius PRESERVED_PLACEHOLDER_^^^^2id:(Nour et al., 8 Mar 2026) OR ti:\2id:(Nour et al., 8 Mar 2026) OR ti:\2query2^^^^ (&&&^^^^2query2^^^^&&&).