---
title: Union-Ball Formulation in Geometry
url: https://www.emergentmind.com/topics/union-ball-formulation
type: topic
---

# Union-Ball Formulation in Geometry

Searching arXiv for the primary paper and a small set of related papers to ground the article.
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{"query":"\"union-ball formulation\" arXiv","max_results":10}
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 \(\mathbb{R}^n\) and asks whether a nonempty closed set \(S\) 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 [2603.07588].

## 1. Formal definition and geometric setting

The ambient space is \(\mathbb{R}^n\) with the usual inner product \(\langle\cdot,\cdot\rangle\) and norm \(\|\cdot\|\). For \(x\in\mathbb{R}^n\) and \(r>0\), \(\overline{B}(x;r)\) denotes the closed ball, \(B(x;r)\) the open ball, and \(\mathbb{S}(x;r)\) the sphere. For a set \(A\subset\mathbb{R}^n\), the notation \(\operatorname{int}A\), \(\operatorname{cl}A\), \(\operatorname{bdry}A\), and \(A^c\) refers respectively to the interior, closure, boundary, and complement; the paper also uses
\[
A' := (\operatorname{int}A)^c = \operatorname{cl}(A^c).
\]

A nonempty closed set \(S\subset\mathbb{R}^n\) satisfies the **interior \(r\)-sphere condition** if for every \(a\in\operatorname{bdry}S\) there exists \(x\in S\) such that
\[
a\in\overline{B}(x;r)\subset S.
\]
This is a local rolling-ball condition from the inside. The paper records an equivalent proximal-normal formulation: \(S\) is regular closed, and for each \(a\in\operatorname{bdry}S\) there exists a unit proximal normal \(\zeta\in N_{S'}^P(a)\) realized by an \(r\)-sphere, equivalently
\[
B\!\left(a+r\frac{\zeta}{\|\zeta\|};\,r\right)\cap S'=\emptyset
\quad\Longleftrightarrow\quad
\overline{B}\!\left(a+r\frac{\zeta}{\|\zeta\|};\,r\right)\subset S.
\]

A set \(S\) is a **union of uniform closed balls** of radius \(\rho>0\) if
\[
S=\bigcup_{\alpha\in I}\overline{B}(x_\alpha;\rho)
\]
for some index set \(I\). The radius is common to all balls; there is no restriction on cardinality, overlap, or center placement beyond exact reconstruction of \(S\). In this language, the union-ball formulation asks for a radius \(r'\) depending only on \(r\) and the dimension such that
\[
S=\bigcup\left\{\overline{B}(x;r'):\ \overline{B}(x;r')\subset S\right\}=S_{r'}.
\]
This is an exact structure theorem, not a volumetric or extremal inequality [2603.07588].

## 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 \(r'>0\). As summarized in the planar paper, a general theorem already shows that one may always take \(r/2\) 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
\[
R_n=\frac{nr}{2\sqrt{n^2-1}},
\]
motivated by examples showing that, in dimension \(n\), radii \(r'>R_n\) fail in general [2603.07588].

| Statement | Radius in the conclusion | Status |
|---|---:|---|
| Weak version | some \(r'>0\) | subsumed by the \(r/2\) theorem |
| General sharp theorem | \(r/2\) | valid in all dimensions |
| Strong version | \(\dfrac{nr}{2\sqrt{n^2-1}}\) | proved in the plane |
| Planar value | \(\dfrac{r}{\sqrt{3}}\) | optimal in \(\mathbb{R}^2\) |

For \(n=2\), the strong radius becomes
\[
R_2=\frac{2r}{2\sqrt{3}}=\frac{r}{\sqrt{3}}.
\]
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 \(r\). The conclusion is global: every point of \(S\) lies in some interior ball of a single radius \(r'\). The quantities singled out in the paper are the boundary distance
\[
d(x,\operatorname{bdry}S)=\min_{a\in\operatorname{bdry}S}\|x-a\|
\]
for \(x\in S\), and the configuration of proximal normals \(N_{S'}^P(a)\) 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 \(r\), then every point of \(S\) lies in an interior ball of radius
\[
R_n=\frac{nr}{2\sqrt{n^2-1}},
\]
and larger uniform radii cannot generally work in dimension \(n\).

The paper also places the formulation in the language of nonsmooth analysis. The interior sphere condition is equivalent to proximal smoothness, and the representation \(S=S_{r'}\) 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 [2603.07588].

## 4. The planar theorem and the proof architecture

The main theorem states that if \(S\subset\mathbb{R}^2\) is nonempty, closed, and satisfies the interior \(r\)-sphere condition, then
\[
S=\bigcup\{\overline{B}(x;r/\sqrt{3}) : \overline{B}(x;r/\sqrt{3})\subset S\}.
\]
Equivalently, \(S=S_{r/\sqrt{3}}\). This proves the strong version in the plane with the optimal constant [2603.07588].

The proof is formulated entirely in union-ball terms. For a candidate radius \(r^*\in(0,r)\), define
\[
S_{r^*}:=\bigcup\{\overline{B}(x;r^*):\ \overline{B}(x;r^*)\subset S\}.
\]
One argues by contradiction, assuming \(S\neq S_{r^*}\). A point \(x_0\in S\setminus S_{r^*}\) is chosen, together with a nearest boundary point \(s_0\in\operatorname{proj}_{\operatorname{bdry}S}(x_0)\), and
\[
r_0=\|x_0-s_0\|,\qquad
\zeta=\frac{x_0-s_0}{\|x_0-s_0\|}.
\]
Since \(x_0\) is not covered by an interior ball of the target radius, one obtains \(r_0<r\).

A core local result, Lemma 3.2, shows that \(s_0\) is non-regular: the proximal normal cone is not a single ray. More precisely, there exist two distinct unit proximal normals \(\zeta_0,\xi_0\in N_{S'}^P(s_0)\), both realized by radius-\(r\) balls, and they satisfy a system of angular inequalities. Their effect is to force a wide cone of inward normals at \(s_0\), together with a tightly constrained location of the direction \(\zeta\) toward \(x_0\). The proof then performs an extremal-ball construction along the ray \(s_0+\tau\zeta\), producing further non-regular boundary points \(s_0',s_0''\) on the boundary of a common interior ball.

The distinctly planar ingredient is Lemma 3.1, a purely two-dimensional geometric lemma about one small circle and two equal larger circles. If
\[
r_0<\frac{r}{\sqrt{3}},
\]
then certain pairs of points on the small circle that avoid both large circles subtend an angle strictly less than \(\pi/3\). After translating this configuration back to the boundary geometry of \(S\), the argument yields
\[
\angle s''_0 s_0 s'_0 < \frac{\pi}{3},\qquad
\angle s_0 s'_0 s''_0 < \frac{\pi}{3},\qquad
\angle s'_0 s''_0 s_0 < \frac{\pi}{3}.
\]
Summing gives a total strictly less than \(\pi\), contradicting the planar triangle angle sum. Hence no such \(x_0\) can exist, and \(S=S_{r/\sqrt{3}}\).

The proof is therefore not merely existential: it identifies the obstruction to coverage by radius-\(r/\sqrt{3}\) 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 [2603.07588].

## 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-\(r\) tangent disks forces global representability as a union of radius-\(r/\sqrt{3}\) disks. Since the constant agrees with the dimension-dependent candidate \(R_n\) at \(n=2\), the theorem is optimal.

From the viewpoint of proximal and variational analysis, the result refines the internal geometry of proximally smooth sets in \(\mathbb{R}^2\). 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 \(\mathbb{R}^n\), one expects an interior ball whose boundary contains at least \(n+1\) 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 \(n\ge 3\), and the paper states that no proof or counterexample is known in those dimensions [2603.07588].

## 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
\[
\Omega=\bigcup_{j=1}^s B(C_j,r_j)\subset\mathbb{R}^3,
\]
with QMC sampling on the solid and on the true boundary
\[
\partial\Omega=\bigcup_{j=1}^s \partial B(C_j,r_j)\,\backslash\,\bigcup_{k=1}^s \overset{\circ}{B}(C_k,r_k).
\]
That formulation supports Tchakaloff-like compression of QMC volume and surface integration on unions of balls [2303.01460].

In topological data analysis, the **localized union-of-balls bifiltration** fixes a center \(q\) and studies
\[
L_{s,r}=\left(\bigcup_{p\in P} B_s(p)\right)\cap B_r(q),
\]
together with a relative variant \(\big(L_s,L_s\setminus B_r^\circ(q)\big)\). Here the phrase refers to a two-parameter refinement of the classical union-of-balls filtration used in persistent homology [2303.07002].

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 [1804.06699].

In adversarial robustness for piecewise-linear neural networks, the relevant geometry is not a union of Euclidean balls but the radius of the largest \(\ell_p\)-ball centered at \(x_0\) 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 [1903.08778].

A complementary notion appears in the study of ball-polyhedra, where one takes finite **intersections** of congruent balls rather than unions:
\[
P=\bigcap_{i=1}^n B(c_i,1).
\]
That literature emphasizes the duality between unions and intersections under complements and studies rigidity of the resulting intersection-of-balls structures [1110.2535].

Finally, in abstract ball spaces, “closure under unions of chains” defines a different order-theoretic completion:
\[
\mathrm{cu}(\mathcal B)=\left\{\bigcup\mathcal C:\ \emptyset\neq\mathcal C\subseteq\mathcal B,\ \mathcal C\text{ is a chain}\right\}.
\]
There the issue is not Euclidean representation by equal-radius balls but stability of spherical completeness under chain-union closure [2404.03597].

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 \(S=S_{r'}\) for sets satisfying a uniform interior sphere condition, with the planar strong theorem now settled at radius \(r/\sqrt{3}\) [2603.07588].

Source: https://www.emergentmind.com/topics/union-ball-formulation