The union of uniform closed balls conjecture (2011) asks whether the union of equal-radius closed balls in Euclidean space has specific regularity properties — properties that constrain how the boundary of the union behaves geometrically. The strong version imposes additional conditions linking proximal normals to the structure of the boundary.
In the plane, the strong conjecture holds (arXiv:2603.07588). The proof is dimension-specific: two-dimensional geometry provides constraints that may not generalize. The conjecture remains open in higher dimensions.
The structural observation: geometric conjectures about “all dimensions” often depend on dimension-specific arguments that reveal different mechanisms at work in different spaces. The planar proof succeeds because the boundary of a union of disks in two dimensions has topological simplicity that the boundary of a union of balls in three dimensions lacks. Proving the conjecture in two dimensions does not bring the higher-dimensional case closer — it may even suggest that the higher-dimensional case requires fundamentally different techniques, or that the conjecture is false there. The dimension that yields to proof is not always the one that illuminates the obstacle.