A counterexample to the strong version of the union of uniform closed balls conjecture in dimensions three and higher

Fifteen years after its formulation and following its recent proof in the plane, we show that the strong version of the union of uniform closed balls conjecture fails in every dimension n≥ 3. For every n≥ 3, $r>0$, and ε>0, we construct a compact set Sⁿ satisfying the interior r-sphere condition and containing a point that belongs to no closed ball of radius (1/2+ε)r contained in S. Consequently, the known universal radius $r/2$ is optimal in every dimension n≥ 3, even for compact sets, whereas the optimal radius in the plane is r/√3. The construction uses simplices with coincident circumcenter and incenter and yields an exact formula for the largest radius of a contained ball covering the center; the regular simplex recovers the original example motivating the conjecture.